![“master” — 2012/6/18 — 10:53 — page 6 — #16
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6 1. Axioms for the Reals
What is a proof?
Originally, to prove something meant to try it out, or to test it
(leaving open, of course, the possibility of failure). For example, a
baker proofs the yeast before baking, and Christopher Marlowe’s pas-
sionate shepherd says, “Come live with me and be my love, and we
will all the pleasures prove.”
In mathematics, on the other hand, a proof is supposed to be an ar-
gument that leaves no possible doubt of its truth. Mathematical proofs
are not supposed to fail. Some do, however, because of human error,
because standards of proof evolve so that what was once certain be-
comes doubtful or even wrong, and because of the intrinsic subtlety
of the whole process.
No one has ever set down exactly what makes a mathematical
proof a proof. No one knows infallibly what a proof is. On the other
hand, attacking proofs is easy: every statement that does not carry
utter conviction is vulnerable to criticism.
Perhaps proof, like Zen Buddhism (as described by Alan Watts)
“can have no positive definition. It has to be suggested by saying what
it is not, somewhat like a sculptor reveals an image by the act of re-
moving pieces of stone from a block.” Unfortunately, this makes find-
ing proofs something like “a game in which the rules [have] been
partially concealed.”
How do we cope with this situation? When a proof is proposed, it
is read by other mathematicians. As more and more people study it,
test it, and work with it, it gradually achieves acceptance.
In summary, although we can’t say exactly what a proof is, it’s
always possible to proof proofs, that is, to test them. To test your
proof, let others read it. See whether they are convinced.
Problems
8. Describe a typical equivalence class for the equivalence relations de-
scribed in problems 1–5.
(Example: For congruence for triangles (Problem 1), a typical equiva-
lence class consists of all triangles of a particular shape and size.)](https://image.slidesharecdn.com/25996206-250524065938-bb16f5c0/85/Which-Numbers-Are-Real-Classroom-Resource-Materials-Michael-Henle-21-320.jpg)
![“master” — 2012/6/18 — 10:53 — page 12 — #22
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12 1. Axioms for the Reals
quaternions in order to calculate with vectors and compute transformations
of three-dimensional space, the constructive reals in order to deal with per-
ceived limits on the nature of computation, the hyperreals in order to cal-
culate with infinitesimals, and the surreals in order to evaluate positions in
combinatorial games. Their discovery is a legacy of theoretical develop-
ments of the 19th century including the description of the rationals as F=.
For more on the history of number systems, see references [A1] and [A2].
For more on the history of mathematics in general, see [A3].
Summary
Number systems have a long practical history but their theory developed
only rather recently. One important theoretical development is the inven-
tion of equivalence relations, which permit the elements of a set to serve
as names for a new set constructed from the old. Equivalence relations ap-
pear in many parts of mathematics. In this book they are used to create new
number systems.
1.2 The Field Axioms
The algebra of the reals
In this and the next two sections we describe a categorical axiom system for
the reals. The axioms of this system divide conveniently into three subsets:
axioms for the algebra of the reals, axioms for the geometry of the reals,
and a final axiom called completeness.
Algebra is the most familiar of these; it is the natural place to start.
Definition. Let S be a set with two operations called addition and mul-
tiplication and written with the usual signs. We assume that the set S is
closed under these operations, so that applying the operations to two ele-
ments of S produces another element of S.
Then S is a field if addition and multiplication have the properties:
(a) For a, b, c in S,
.aCb/Cc D aC.bCc/, — associativity of addition
and
.ab/c D a.bc/. — associativity of multiplication
(b) For a, b in S,](https://image.slidesharecdn.com/25996206-250524065938-bb16f5c0/85/Which-Numbers-Are-Real-Classroom-Resource-Materials-Michael-Henle-27-320.jpg)
![“master” — 2012/6/18 — 10:53 — page 13 — #23
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1.2. The Field Axioms 13
a C b D b C a, — commutativity of addition
and
ab D ba. — commutativity of multiplication
(c) For a, b in S,
a.b C c/ D ab C ac. — distributivity
(d) There are distinct, special elements 0 (zero) and 1 (one) in S such that
for any b in S,
0 C b D b C 0 D b, — additive identity
and
1b D b1 D b. — multiplicative identity
(e) For any b in S, there is an element b in S so that
b C . b/ D 0, — additive inverse
and for any b in S, except 0, there is an element c in S so that
bc D 1, — multiplicative inverse
Properties (a)–(e) are the field axioms. They summarize the algebra of
the real numbers, the rational numbers, and many other number systems.
They were first collected together under the term ‘field’ by Weber in 1893.
See reference [C7].
Assumption. Let Q be the set of all rational numbers, that is numbers of
the form p=q, where p and q are integers and q is not zero. We assume that
Q is a field.
Problems
1. Let Z be the set of all integers. Z is not a field. List the axiom(s) not
satisfied by Z.
2. Let n be a positive integer, and consider the set Z=n of integers modulo
n, that is Z=n is the set of equivalence classes of Z under the equiva-
lence relation x y .mod n/ if n divides x y with remainder zero.
How many equivalence classes are there in Z=n?
3. Show that addition and multiplication are well-defined on Z=n, that is
if a b .mod n/ and c d .mod n/, then a C c b C d .mod n/
and ac bd .mod n/.](https://image.slidesharecdn.com/25996206-250524065938-bb16f5c0/85/Which-Numbers-Are-Real-Classroom-Resource-Materials-Michael-Henle-28-320.jpg)
![“master” — 2012/6/18 — 10:53 — page 31 — #41
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1.4. The Completeness Axiom 31
[ ,
a1
[ ,
a2
[ ,
a3 b3]
b4]
[ ,
a4
S
G
b1]
b2]
Figure 1.4.3. Proof by bisection: the first few intervals.
Completeness = Cauchy Completeness + the
Archimedean property
The Archimedean property and Cauchy completeness are consequences of
order completeness. Conversely, together they imply order completeness.
This is useful as it allows two simpler ideas to replace the difficult con-
cept of order completeness. The proof of this theorem is also significant. It
employs a powerful technique called proof by bisection.
Theorem 1.4.3. An ordered field that is Archimedean and Cauchy complete
is order complete.
Proof. Let S be a linearly ordered field that is both Archimedean and Cauchy
complete. Let G be a non-empty subset of S that is bounded above. Our goal
is to prove that G has a least upper bound.
The proof uses a trick; a trick so useful that it has become a technique:
proof by bisection. We start with an interval Œa1; b1, which is then divided
in half at its midpoint .a1 Cb1/=2. One of the resulting half intervals is cho-
sen, call it Œa2; b2, and subdivided in turn into subintervals that are quarters
of the original interval. One of these subintervals, Œa3; b3, is chosen and in
turn divided in half and one of those halves chosen. Continuing to choose
subintervals and halve them, the ith interval obtained is called Œai; bi . This
is the setup for proof by bisection. It is illustrated in Figure 1.4.3.
What we have not yet explained is how to pick the very first interval
Œa1; b1 and how to choose between the two subintervals at each step be-
fore proceeding to the next bisection. These choices depend on what one
is trying to prove. However, in all proofs by bisection the left endpoints ai
increase and the right endpoints bi decrease. In the middle is a point c that
is the limit of both sequences of endpoints. Under favorable circumstances,
c has whatever property is needed to complete the proof.
For the proof of Theorem 1.4.3 we choose Œai ; bi so that bi is an upper
bound for G and ai is not. (See Figure 1.4.3.) The proof is completed by](https://image.slidesharecdn.com/25996206-250524065938-bb16f5c0/85/Which-Numbers-Are-Real-Classroom-Resource-Materials-Michael-Henle-46-320.jpg)
![Nach V e r e d a r i u s [30] repräsentierten 1884 die oberirdischen
u n d versenkten Linien z u s a m m e n eine Länge von rund 1200000
km mit 3650000 km Leitungsdrähten. Letztere würden also
hinreichen, n e u n z i g m a l den Gleichen zu umspannen.
Der G e s a m t a u f w a n d a n K a p i t a l für Land- und
Seetelegraphen betrug nach M u l h a l l bis zum Dezember 1882 rund
88 Mill. Pfd. St. = 1760 Mill. Mk.
4. Gebührentarif für Telegramme von Stationen
des Deutschen Reiches[31]
.
(Für den billigsten und gebräuchlichsten Weg berechnet.)
A. Die Wortlänge ist festgesetzt auf 15
Buchstaben oder 5 Ziffern im Verkehr mit:
Wort-
Taxe
Mark.
Deutschland (innerer Verkehr) 0,06
Afrika (West-): Kanarische Inseln 1,45
Senegal 2,65
Bolama 5,85
Bissao u. Konakry 5,90
Algerien-Tunis 0,27
Belgien 0,10
Bosnien-Herzegowina 0,20
Bulgarien 0,25
Dänemark 0,10
Frankreich 0,15
Gibraltar 0,25
Griechenland
a) Festland und Insel Paros 0,40
b) nach den übrigen Inseln 0,45
Großbritannien und Irland 0,20
Außerdem ist für jedes Telegramm nach
Großbritannien u. Irland eine Grundtaxe
von 0,40 Mark zu erheben.
Helgoland 0,15](https://image.slidesharecdn.com/25996206-250524065938-bb16f5c0/85/Which-Numbers-Are-Real-Classroom-Resource-Materials-Michael-Henle-56-320.jpg)
![Anhang.
Das Fernsprechwesen[32].
1. G e s c h i c h t e . Die Versuche, den Schall mittels geeigneter
Übertragung der Schallwellen fortzuleiten, gehören schon einer
ziemlich weit zurückliegenden Vergangenheit an. So weist der
englische Elektriker P r e e c e nach, daß sein Landsmann, der
Physiker R o b e r t H o o k e , bereits 1667 derartige, wenn auch noch
ziemlich rohe Versuche anstellte, indem derselbe einen
ausgespannten Faden benutzte. Einen telephonischen Apparat
konstruierte auch W h e a t s t o n e im Jahre 1819. Aber erst 1861
fertigte der 1874 verstorbene Lehrer P h i l i p p R e i s in
Friedrichsdorf bei Frankfurt a. M. das e r s t e e l e k t r i s c h e
Te l e p h o n . Dieser von Reis mit dem von ihm selbst erfundenen
Worte „Te l e p h o n “[33] bezeichnete Apparat übertrug musikalische
Töne und Melodieen, ferner auch Worte, wenn schon in etwas
unvollkommener Weise, auf ziemlich weite Entfernungen. Die ganze
Sache wurde indes von den Physikern nur als eine Kuriosität, nicht
als praktisch wichtig betrachtet[34], und auch Reis selbst hatte seinen
Apparat von Anfang an nur für Unterrichtszwecke bestimmt. So kam
es, daß der deutsche Erfinder und sein Instrument in Europa nach
kurzer Zeit wieder vergessen wurden. In Amerika dagegen wurde der
deutsche Gedanke weiter verfolgt. 1868 konstruierte dort ein
gewisser v a n d e r W e y d e ein verbessertes Reissches Telephon,
das deutlich, wenn auch nur schwach und mit näselndem Tone,
hineingesprochene Worte übertragen haben soll. V a n d e r
W e y d e setzte seine Versuche fort, und seinen Bestrebungen schloß
sich E l i s h a G r a y in Chicago an. Aber all diese Telephone, wie
auch die in England gefertigten, eigneten sich in der Hauptsache nur](https://image.slidesharecdn.com/25996206-250524065938-bb16f5c0/85/Which-Numbers-Are-Real-Classroom-Resource-Materials-Michael-Henle-63-320.jpg)
![zur Übertragung musikalischer Töne, nicht aber für artikulierten
Schall, d. i. für die Wiedergabe der Sprache. Dieses so schwierige
Problem wurde durch den Taubstummenlehrer G r a h a m B e l l in
B o s t o n , einen geborenen Schotten, im Jahre 1876 glücklich gelöst
und so die Welt von Amerika her mit dem p r a k t i s c h e n Telephon
beschenkt. Seitdem gelang es, durch verschiedene Verbesserungen
die telephonische Wirkung bedeutend zu erhöhen und überhaupt
den Fernsprech-Apparat für die Verwendung im Verkehre noch
bequemer zu gestalten. Großartiges zeigte bezüglich des
Fernsprechwesens besonders die internationale elektrische
Ausstellung zu Philadelphia im Jahre 1884. Der dort ausgestellte
Quadruplex-Translator E d i s o n s z. B. verstärkte den Ton vierfach;
sein Mikrophon[35] ließ den Schritt einer Fliege deutlich hören; das
größte Aufsehen aber erregte unter den Laien sein lautsprechendes
Telephon, dessen Töne im Umkreis von 30 Fuß deutlich vernehmbar
waren, und dessen hohe Noten bedeutend ausgeprägter waren als
die tieferen[36]. Sicher wird auch d i e Zeit nicht ausbleiben, wo man,
wie schon R e i s andeutete, die menschliche Stimme übers Meer
senden wird, wie das mittels des Telegraphen bezüglich der Schrift
bereits der Fall ist[37].](https://image.slidesharecdn.com/25996206-250524065938-bb16f5c0/85/Which-Numbers-Are-Real-Classroom-Resource-Materials-Michael-Henle-64-320.jpg)
![Fig. 15. Philipp Reis.
In Deutschland wurde das erste Fernsprechamt für den
öffentlichen Verkehr am 12. November 1877 in Friedrichsberg bei
Berlin eröffnet, und heute (Ende 1885) giebt es, dank der Thatkraft
des obersten Leiters der deutschen Reichspost- und
Telegraphenverwaltung, D r. v o n S t e p h a n s , in 81 Orten 12655
Fernsprechstellen und 21357 km Drahtleitungen[38].
Auch in den übrigen Kulturländern der Erde hat das
Fernsprechwesen fast überall Eingang gefunden; selbst das Reich der
Mitte hat sich nicht ausschließen können. Shanghai zählt bereits 77,](https://image.slidesharecdn.com/25996206-250524065938-bb16f5c0/85/Which-Numbers-Are-Real-Classroom-Resource-Materials-Michael-Henle-65-320.jpg)
![Hongkong 40 Fernsprechstellen; ja sogar die Hauptstadt der
Sandwich-Inseln, Honolulu, hat ihre Telephonleitung[39].
2. R e c h t s v e r h ä l t n i s s e . Die Rechtsverhältnisse im
Fernsprechbetrieb sind sehr verschiedenartig. Ganz frei in
Anwendung und Ausbeutung ist der Betrieb in den Vereinigten
Staaten, in Schweden, Norwegen und in den meisten Kolonieen;
g a n z v o m S t a a t a b h ä n g i g i m D e u t s c h e n R e i c h
u n d i n d e r S c h w e i z ; unter der Kontrolle der Regierung in
England, Rußland, Österreich, Frankreich, Italien, Spanien und
Portugal. Doch ist in England, Rußland und Österreich der Betrieb
den Privatgesellschaften auf l a n g e Zeit vertragsmäßig gewährt, in
den anderen nur auf k u r z e Fristen.
3. S t a t i s t i s c h e s . Ein annäherndes Bild von der Verbreitung
des Fernsprechers in der Mehrzahl der europäischen Staaten im Jahr
1885 giebt folgende Tabelle[40]:
Länder.
Städte
mit
Fernsprech-
Einrichtungen.
Abgerundete
Zahl der
Stellen.
Jährlicher
Abonnements-
betrag in
Mark.
Deutschland 81 13000 150
England 180 (?) 12000 100–400
Frankreich etwa 20 10000 480
Italien 18 7000 92–140
Schweden 51 10000 128–216
Schweiz 30 5000 120–200
Spanien unbekannt 1000 80–200
Niederlande etwa 11 4000 136–204
Belgien 12 5000 160–200
Rußland 7 3000 560
Österreich-Ungarn 10 4500 180–300
In den V e r e i n i g t e n S t a a t e n von Amerika betrug 1884 die
Länge der Telephonlinien 193120 km (vgl. Gothaischer Kalender für
1886). Im gleichen Jahre zählte N e w -Yo r k mit Umgebung schon
10600 Abonnenten, während gleichzeitig ganz England nur 11000–](https://image.slidesharecdn.com/25996206-250524065938-bb16f5c0/85/Which-Numbers-Are-Real-Classroom-Resource-Materials-Michael-Henle-66-320.jpg)
![II.
Weltpost.
Erstes Kapitel.
Geschichte des Postwesens[41].
I. Altertum.
Die Staaten als solche, d. h. die Regierungen, hatten schon
frühzeitig für ihre Zwecke bestimmte Anstalten zur Herstellung
gesicherter und schneller Verbindungen errichtet. Dabei wurden
anfänglich die im Dienste des Herrschers stehenden Boten von der
Hauptstadt aus mit den Befehlen an die obersten Verwaltungschefs,
die Truppenbefehlshaber u. s. w. in den Provinzen d i r e k t
abgesandt, und sie brachten die Berichte auch wieder zurück. Sehr
bald aber kam man auf den Gedanken der Errichtung von Stationen
und des stationsweisen Transportes mittels Wechsels des
Beförderungsmittels, wodurch zugleich eine erhebliche
Beschleunigung erzielt wurde. Solche B o t e n a n s t a l t e n besaßen
bereits die Regierungen in I n d i e n , C h i n a , Ä g y p t e n ,
A s s y r i e n , B a b y l o n i e n und die K ö n i g e d e r H e b r ä e r .
In I n d i e n waren an den Endpunkten der ziemlich kurzen
Stationen Hütten errichtet. Sobald ein Bote bei einer solchen Hütte
ankam, empfing der schon bereitstehende andere das Schreiben, um](https://image.slidesharecdn.com/25996206-250524065938-bb16f5c0/85/Which-Numbers-Are-Real-Classroom-Resource-Materials-Michael-Henle-69-320.jpg)
![Straßen freilich nur für militärische Zwecke bestimmt; aber sie
dienten doch schon frühzeitig auch dem V e r k e h r e . So gingen vor
allem, wie in den übrigen älteren Reichen, s t a a t l i c h e B o t e n
von Rom zu den auswärts bestellten Beamten und Befehlshabern,
um Befehle oder Nachrichten zu überbringen, oder es wurden von
diesen solche nach Rom gesendet. Die Boten hießen viatores,
cursores, statores, tabellarii (letzterer Name rührt davon her, daß die
Alten statt der Briefbogen Täfelchen [tabellae] benutzten). Die
Vergütung, welche sie für die Übermittlung von Nachrichten
erhielten, nannte man calcearium, Schuhgeld[42].
Eine bedeutende Förderung wurde dem Nachrichten- und auch
Frachtenverkehr zu teil durch jene große Gesellschaft römischer
Ritter, welche in den letzten Zeiten der Republik die Staatsländereien
in den Provinzen, sowie die Zehnten, Gefälle und Steuern pachtete
und einen ausgedehnten, schwunghaften Handel mit Getreide und
anderen Landesprodukten betrieb. Diese Genossenschaft hatte ihren
Centralsitz in Rom und ihre Niederlagen und Comptoire in allen
wichtigeren Provinzstädten. Ihr Nachrichten- und Geldverkehr vom
Mittelpunkte nach den Filialen und zwischen diesen selber wieder
war ein großartiger, und deshalb unterhielt die Gesellschaft eine
große Zahl von Briefträgern (tabellarii), welche Briefe und leichtes
Gepäck bis in die kleinsten Städte aller Provinzen mit großer
Schnelligkeit und ziemlicher Regelmäßigkeit beförderten. Diese
Briefträger durften auch Sendungen von Privaten übernehmen und
wurden häufig hierzu benützt.
Außerdem gab es noch zahlreiche Privatboten. Reiche Familien,
die in Rom wohnten, hatten große Güter in den Provinzen, oder ihre
Söhne studierten an griechischen Schulen. Da sie nun mit ihren
Verwaltern und ihren Kindern in regelmäßigem Verkehre bleiben
wollten, so unterhielten sie Briefboten, die nicht bloß von ihnen,
sondern auch von Bekannten mit Sendungen betraut wurden.
Häufig wurden auch Reisenden, Schiffern, Kaufleuten, Fuhrleuten
u. s. w. Briefschaften zur Abgabe in den Orten, wohin ihr Geschäft
sie führte, übergeben. Freilich war diese Art der Beförderung in](https://image.slidesharecdn.com/25996206-250524065938-bb16f5c0/85/Which-Numbers-Are-Real-Classroom-Resource-Materials-Michael-Henle-74-320.jpg)
![Fig. 17. Die Staatspost unter den römischen Kaisern.
(Nach dem „Poststammbuch“.)
Das größte Verdienst in dieser Beziehung erwarb sich gleich der
erste römische Imperator, O c t a v i a n u s A u g u s t u s , durch die
Errichtung des sogen. cursus publicus.
Der cursus publicus war eine S t a a t s v e r k e h r s a n s t a l t ,
welche die Beförderungen stationsweise, mit Wechsel der
Transportmittel, zu Fuß, zu Pferd oder zu Wagen sowohl für
V e r s e n d u n g e n , als auch für R e i s e n wahrzunehmen hatte.
Diese Einrichtung war zunächst bestimmt für die Reisen des Kaisers
und seines Hofes, dann der Militärpersonen und Staatsbeamten im
Dienste, der Gesandten und der zur Benutzung des cursus publicus
im einzelnen Falle besonders ermächtigten Personen[43]; ferner zur
Beförderung der Depeschen, Akten, Dokumente und der](https://image.slidesharecdn.com/25996206-250524065938-bb16f5c0/85/Which-Numbers-Are-Real-Classroom-Resource-Materials-Michael-Henle-76-320.jpg)
Which Numbers Are Real Classroom Resource Materials Michael Henle
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- 5. THE SURREAL NUMBERS THE HYPERREAL NUMBERS THE COMPLEX NUMBERS THE QUATERNIONS THE CONSTRUCTIVE REALS Which Numbers are Real? Michael Henle CLASSROOM RESOURCE MATERIALS CLASSROOM RESOURCE MATERIALS Which Numbers are Real? surveys alternative real number systems: systems that generalize and extend the real numbers while staying close to the properties that make the reals central to mathematics. These systems include, for example, multi-dimensional numbers (the complex numbers, the quaternions, and others), systems that include infinitely small and infinitely large numbers (the hyperreal numbers and the surreal numbers), and numbers that represent positions in games (the surreal numbers). All the systems presented have applications and several are the subject of current mathematical research. Which Numbers are Real? Which Numbers are Real? Michael Henle Michael Henle Which Numbers are Real? will be of interest to anyone who likes numbers, but particularly upper-level undergraduates, graduate students, and mathematics teachers at all levels. 9 780883 857779 ISBN 978-0-88385-777-9 MAA MAA-Real Numbers cover v9_Layout 1 6/19/12 1:34 PM Page 1
- 6. “master” — 2012/6/18 — 10:53 — page i — #1 i i i i i i i i Which Numbers are Real?
- 7. “master” — 2012/6/18 — 10:53 — page ii — #2 i i i i i i i i c 2012 by the Mathematical Association of America, Inc. Library of Congress Catalog Card Number 2012937493 Print edition ISBN 978-0-88385-777-9 Electronic edition ISBN 978-1-61444-107-6 Printed in the United States of America Current Printing (last digit): 10 9 8 7 6 5 4 3 2 1
- 8. “master” — 2012/6/18 — 10:53 — page iii — #3 i i i i i i i i Which Numbers are Real? Michael Henle Oberlin College Published and Distributed by The Mathematical Association of America
- 9. “master” — 2012/6/18 — 10:53 — page iv — #4 i i i i i i i i Council on Publications and Communications Frank Farris, Chair Committee on Books Gerald M. Bryce, Chair Classroom Resource Materials Editorial Board Gerald M. Bryce, Editor Michael Bardzell Jennifer Bergner Diane L. Herrmann Philip P. Mummert Barbara E. Reynolds Susan G. Staples Philip D. Straffin Cynthia J Woodburn Holly S. Zullo
- 10. “master” — 2012/6/18 — 10:53 — page v — #5 i i i i i i i i CLASSROOM RESOURCE MATERIALS Classroom Resource Materials is intended to provide supplementary class- room material for students—laboratory exercises, projects, historical in- formation, textbooks with unusual approaches for presenting mathematical ideas, career information, etc. 101 Careers in Mathematics, 2nd edition edited by Andrew Sterrett Archimedes: What Did He Do Besides Cry Eureka?, Sherman Stein Calculus:An Active Approach with Projects, Stephen Hilbert, Diane Driscoll Schwartz, Stan Seltzer, John Maceli, and Eric Robinson The Calculus Collection: A Resource for AP and Beyond, edited by Caren L. Diefenderfer and Roger B. Nelsen Calculus Mysteries and Thrillers, R. Grant Woods Conjecture and Proof, Miklós Laczkovich Counterexamples in Calculus, Sergiy Klymchuk Creative Mathematics, H. S. Wall Environmental Mathematics in the Classroom, edited by B. A. Fusaro and P. C. Kenschaft Excursions in Classical Analysis: Pathways to Advanced Problem Solving and Undergraduate Research, by Hongwei Chen Exploratory Examples for Real Analysis, Joanne E. Snow and Kirk E. Weller Geometry From Africa: Mathematical and EducationalExplorations, Paulus Gerdes Historical Modules for the Teaching and Learning of Mathematics (CD), edited by Victor Katz and Karen Dee Michalowicz Identification Numbers and Check Digit Schemes, Joseph Kirtland Interdisciplinary Lively Application Projects, edited by Chris Arney Inverse Problems: Activities for Undergraduates, Charles W. Groetsch Laboratory Experiences in Group Theory, Ellen Maycock Parker Learn from the Masters, Frank Swetz, John Fauvel, Otto Bekken, Bengt Johansson, and Victor Katz Math Made Visual: Creating Images for UnderstandingMathematics, Claudi Alsina and Roger B. Nelsen Mathematics Galore!: The First Five Years of the St. Marks Institute of Mathematics, James Tanton
- 11. “master” — 2012/6/18 — 10:53 — page vi — #6 i i i i i i i i Ordinary Differential Equations: A Brief Eclectic Tour, David A. Sánchez Oval Track and Other Permutation Puzzles, John O. Kiltinen A Primer of Abstract Mathematics, Robert B. Ash Proofs Without Words, Roger B. Nelsen Proofs Without Words II, Roger B. Nelsen Rediscovering Mathematics: You Do the Math, Shai Simonson She Does Math!, edited by Marla Parker Solve This: Math Activities for Students and Clubs, James S. Tanton Student Manual for Mathematics for Business Decisions Part 1: Probabil- ity and Simulation, David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic Student Manual for Mathematics for Business Decisions Part 2: Calculus and Optimization, David Williamson, Marilou Mendel, Julie Tarr, and Deborah Yoklic Teaching Statistics Using Baseball, Jim Albert Visual Group Theory, Nathan C. Carter Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go, Annalisa Crannell, Gavin LaRose, Thomas Ratliff, Elyn Rykken MAA Service Center P.O. Box 91112 Washington, DC 20090-1112 1-800-331-1MAA FAX: 1-301-206-9789
- 12. “master” — 2012/6/18 — 10:53 — page vii — #7 i i i i i i i i Introduction Alternative reality The real numbers are fundamental. Although mostly taken for granted, they are what make possible all of mathematics from high school algebra and Euclidean geometry through the calculus and beyond, and also serve as the basis for measurement in science, industry, and ordinary life. In this book we study alternative systems of numbers: systems that generalize and ex- tend the reals yet stay close to the fundamental properties that make the reals central to so much mathematics. By an alternative number system we mean a set of objects that can be combined using two operations, addition and multiplication, and that share some significant algebraic and geometric properties with the real numbers. Exactly what these properties are is made clear in Chapter One. We are not concerned with numeration, however. A numeration system is a means of giving names to numbers, for example, the decimal system for writing real numbers. We go beyond numeration to describe number systems that in- clude numbers different from ordinary numbers including multi-dimensional numbers, infinitely small and infinitely large numbers, and numbers that represent positions in games. Although we present some eccentric and relatively unexplored parts of mathematics, each system that we study has a well-developed theory. Each system has applications to other areas of mathematics and science, in par- ticular to physics, the theory of games, multi-dimensional geometry, formal logic, and the philosophy of mathematics. Most of these number systems are active areas of current mathematical research and several were discov- vii
- 13. “master” — 2012/6/18 — 10:53 — page viii — #8 i i i i i i i i viii Introduction ered relatively recently. As a group, they illuminate the central, unifying role of the reals in mathematics. Design of this book This book is designed to encourage readers to participate in the mathemat- ical development themselves. The proofs of many results are either con- tained in problems or depend on results proved in problems. The problems should be read at least, if not worked out. With two exceptions the chapters are independent and can be read in any order. The exceptions are that the first two chapters contain essential back- ground for the rest of the book and that Chapter Four depends somewhat on the proceeding chapter. Use of this book This book presents material that, in addition to being of general interest to mathematics students, is appropriate for an upper level course for under- graduates that can serve as introduction to or sequel to a course in advanced calculus. Alternatively, it can take the place of a course in the foundations of the real number system, or be given as an upper level seminar emphasiz- ing different methods of proof. Prerequisites are standard sophomore level courses: discrete mathematics, multivariable calculus, and linear algebra. A course in advanced calculus or foundations of analysis would also be useful. The goal The goal is to present some interesting, even exotic, mathematics. I hope to convey a sense of the immense freedom available in mathematics, where even in a mundane and well-established area such as the real numbers, al- ternatives are always possible. Acknowledgements This book owes a tremendous debt to previous expositors in this area to whose work this book is closely tied, in particular, to the work of Elwyn Berlekamp, Errett Bishop, John H. Conway, Richard Guy, James Henle, and Eugene Kleinberg.
- 14. “master” — 2012/6/18 — 10:53 — page ix — #9 i i i i i i i i Contents Introduction vii I THE REALS 1 1 Axioms for the Reals 3 1.1 How to Build a Number System . . . . . . . . . . . . . . . 3 1.2 The Field Axioms . . . . . . . . . . . . . . . . . . . . . . 12 1.3 The Order Axioms . . . . . . . . . . . . . . . . . . . . . . 18 1.4 The Completeness Axiom . . . . . . . . . . . . . . . . . . 24 2 Construction of the Reals 35 2.1 Cantor’s Construction . . . . . . . . . . . . . . . . . . . . 36 2.2 Dedekind’s Construction of the Reals . . . . . . . . . . . . 43 2.3 Uniqueness of the Reals . . . . . . . . . . . . . . . . . . . 46 2.4 The Differential Calculus . . . . . . . . . . . . . . . . . . 50 2.5 A Final Word about the Reals . . . . . . . . . . . . . . . . 52 II MULTI-DIMENSIONAL NUMBERS 55 3 The Complex Numbers 57 3.1 Two-Dimensional Algebra and Geometry . . . . . . . . . . 57 3.2 The Polar Form of a Complex Number . . . . . . . . . . . 62 3.3 Uniqueness of the Complex Numbers . . . . . . . . . . . . 66 3.4 Complex Calculus . . . . . . . . . . . . . . . . . . . . . . 70 3.5 A Final Word about the Complexes . . . . . . . . . . . . . 76 ix
- 15. “master” — 2012/6/18 — 10:53 — page x — #10 i i i i i i i i x Contents 4 The Quaternions 77 4.1 Four-Dimensional Algebra and Geometry . . . . . . . . . 77 4.2 The Polar Form of a Quaternion . . . . . . . . . . . . . . . 82 4.3 Complex Quaternions and the Quaternion Calculus . . . . 86 4.4 A Final Word about the Quaternions . . . . . . . . . . . . 94 III ALTERNATIVE LINES 95 5 The Constructive Reals 97 5.1 Constructivist Criticism of Classical Mathematics . . . . . 97 5.2 The Constructivization of Mathematics . . . . . . . . . . . 103 5.3 The Definition of the Constructive Reals . . . . . . . . . . 109 5.4 The Geometry of the Constructive Reals . . . . . . . . . . 114 5.5 Completeness of the Constructive Reals . . . . . . . . . . 118 5.6 The Constructive Calculus . . . . . . . . . . . . . . . . . . 119 5.7 A Final Word about the Constructive Reals . . . . . . . . . 124 6 The Hyperreals 125 6.1 Formal Languages . . . . . . . . . . . . . . . . . . . . . . 126 6.2 A Language for the Hyperreals . . . . . . . . . . . . . . . 134 6.3 Construction of the Hyperreals . . . . . . . . . . . . . . . 138 6.4 The Transfer Principle . . . . . . . . . . . . . . . . . . . . 142 6.5 The Nature of the Hyperreal Line . . . . . . . . . . . . . . 150 6.6 The Hyperreal Calculus . . . . . . . . . . . . . . . . . . . 156 6.7 Construction of an Ultrafilter . . . . . . . . . . . . . . . . 162 6.8 A Final Word about the Hyperreals . . . . . . . . . . . . . 170 7 The Surreals 171 7.1 Combinatorial Games . . . . . . . . . . . . . . . . . . . . 172 7.2 The Preferential Ordering of Games . . . . . . . . . . . . 178 7.3 The Arithmetic of Games . . . . . . . . . . . . . . . . . . 184 7.4 The Surreal Numbers . . . . . . . . . . . . . . . . . . . . 188 7.5 The Nature of the Surreal Line . . . . . . . . . . . . . . . 192 7.6 More Surreal Numbers . . . . . . . . . . . . . . . . . . . 196 7.7 Analyzing Games with Numbers . . . . . . . . . . . . . . 200 7.8 A Final Word about the Surreals . . . . . . . . . . . . . . 204 Bibliography 205 Index 209 About the Author 219
- 16. “master” — 2012/6/18 — 10:53 — page 1 — #11 i i i i i i i i Part I THE REALS What makes a number system a number system? In this book the real numbers serve as the standard with which other number systems are com- pared. To be called a number system a mathematical system must share most if not all of the fundamental properties of the reals. What are the fundamental properties of the reals? We use a set of prop- erties (or laws or axioms) that characterize the reals completely, meaning that any mathematical system with these properties is the same as the re- als. Such a set of properties for a particular mathematical object is called a categorical axiom system. Many such systems are known. A famous one for plane geometry goes back to Euclid, although a correct and complete categorical axiom system for Euclidean geometry was formulated only late in the 19th century. In this later period and on into the 20th century there has been tremendous interest in axiom systems and their application to all areas of mathematics. Part One of this book describes a categorical axiom system for the reals. Chapter One lists the axioms of this system. Chapter Two constructs the reals (from the rational numbers), and shows that they satisfy the axioms presented in Chapter One. In addition, we prove that any mathematical sys- tem satisfying these axioms is identical (more technically, isomorphic) to the reals. A categorical axiom system is a powerful tool. The one we describe is used in this book to analyze and compare number systems. Given any sys- tem we ask: which axioms for the reals does it satisfy? The answer reveals how close the new system is to the standard set by the reals themselves. Of course, any proposed system that is genuinely different from the reals cannot satisfy all the axioms, for then it would be the reals. In principle, we could survey systematically all possible number sys- tems, first by finding those satisfying all but one of the axioms of the reals,
- 17. “master” — 2012/6/18 — 10:53 — page 2 — #12 i i i i i i i i 2 then finding systems satisfying all but two axioms, and so forth. We cannot accomplish this ambitious program; only partial results are known! Even these encompass a veritable ocean of important modern mathematics. How- ever, we can present the most important mathematical systems that satisfy all but a few of the axioms. Important Note: We assume that the reader is already familiar with two kinds of numbers: the integers (i.e., the whole numbers, positive, negative and zero), and the rational numbers (the common fractions). The basic prop- erties of these numbers are assumed. The theory of the reals and other num- ber systems will be based on them.
- 18. “master” — 2012/6/18 — 10:53 — page 3 — #13 i i i i i i i i 1 Axioms for the Reals 1.1 How to Build a Number System Equivalence relations In this section we describe a process that is used to construct many num- ber systems and other mathematical systems as well. It is used to construct about half of the number systems in this book. This process uses the concept of an equivalence relation. Here is the definition: Definition. Let S be a set and a relationship that may or may not hold between two elements of S. A relation on S satisfying these properties: (a) For every a in S, a a, — reflexivity (b) For a, b in S, if a b, then b a, — symmetry (c) For a, b, c in S, if a b, and b c, then a c. — transitivity is called an equivalence relation on S. The three properties—reflexivity, symmetry, and transitivity—are the axioms of equivalence relations, sometimes called laws of equivalence. An example of an equivalence relation is the relation of equality. Every set S has this relation. The idea of an equivalence relation is an abstraction of equality, and, as we will soon see, every equivalence relation can be turned into the relationship of equality on some set. We see many examples of equivalence relations connected with number systems later. Here are a few diverse examples. 3
- 19. “master” — 2012/6/18 — 10:53 — page 4 — #14 i i i i i i i i 4 1. Axioms for the Reals Problems 1. Let S be the set of all triangles in the plane. For A, B in S, set A B if A and B are congruent. Explain why is an equivalence relation for S. (Hint: We are asked to show that “” has the three properties: reflexiv- ity, symmetry and transitivity. For example, is reflexive? According to the definition, this means: Is every triangle congruent to itself? Con- gruence of triangles means that corresponding sides are equal (in some order) and corresponding angles. Therefore, any triangle is congruent to itself: the sides are equal to themselves, and the angles are equal to themselves. The answer is yes: congruence is reflexive.) 2. Consider two further relations between triangles: A B if A and B are similar triangles, A † B if at least one angle of A equals an angle of B. Are and † equivalence relations? 3. Let S be the set of people in Australia. For A, B in S define A B if A and B have the same birthday. Is an equivalence relation? 4. Let a set U be given. Let S be the set of subsets of U . For A, B in S, define A , B if jAj D jBj (i.e., A and B have the same number of elements). Is , an equivalence relation? 5. Let Z be the set of integers, positive, negative and zero. Let p be a positive integer. For x and y in Z define x y .mod p/ if p divides x y with remainder zero. Is an equivalence relation? 6. Let F be the set of all symbols of the form a=b where a and b are integers and b is not zero. (For the purpose of this exercise forget that = is sometimes used for fractions.) Define a relation for these symbols a=b c=d if ad D bc. Is an equivalence relation? 7. Find a set S and a relation on S that is (a) reflexive, but not symmetric or transitive, (b) symmetric, but not reflexive or transitive, (c) transitive, but not reflexive or symmetric, (d) reflexive and symmetric but not transitive, (e) symmetric and transitive but not reflexive,
- 20. “master” — 2012/6/18 — 10:53 — page 5 — #15 i i i i i i i i 1.1. How to Build a Number System 5 (f) transitive and reflexive but not symmetric. (Hint: Good examples of relationships can be found outside mathe- matics. For example, “cousinhood” has an interesting combination of properties. It is symmetric, but not reflexive (i.e., one is not one’s own cousin). Examples of relations can be found on any finite set. Furthermore, if S is finite, a relation on S can be diagrammed as in Figure 1.1.1. The set S there has only three elements, A, B, and C. The arrows indicate the relationships that exist among the elements of S. If the relation is symbolized by “:”, say, then A:B, B:C and C:C. It is not reflexive (A is not related to itself), nor symmetric (A:B, but not B:A), nor transitive (A:B and B:C, but not A:C). A B C Figure 1.1.1. A simple relation on a small set. Equivalence classes To use equivalence relations to construct number systems, we need another definition. Definition. Let S be a set with an equivalence relation . For each a in S let Sa D fbjb ag: The set Sa is called the equivalence class containing a. The set of all the equivalence classes is written S=. Usually, an equivalence relation is defined on a set S precisely in or- der to study the set S= of equivalence classes. S= is the new set built by introducing the equivalence relation. In this book this new set will usu- ally be a number system but the same construction is used for many other mathematical concepts. Proof is one of the most difficult of mathematical concepts (see the shaded text below). The next exercises provide the opportunity to supply proofs of some fundamental facts about equivalence classes.
- 21. “master” — 2012/6/18 — 10:53 — page 6 — #16 i i i i i i i i 6 1. Axioms for the Reals What is a proof? Originally, to prove something meant to try it out, or to test it (leaving open, of course, the possibility of failure). For example, a baker proofs the yeast before baking, and Christopher Marlowe’s pas- sionate shepherd says, “Come live with me and be my love, and we will all the pleasures prove.” In mathematics, on the other hand, a proof is supposed to be an ar- gument that leaves no possible doubt of its truth. Mathematical proofs are not supposed to fail. Some do, however, because of human error, because standards of proof evolve so that what was once certain be- comes doubtful or even wrong, and because of the intrinsic subtlety of the whole process. No one has ever set down exactly what makes a mathematical proof a proof. No one knows infallibly what a proof is. On the other hand, attacking proofs is easy: every statement that does not carry utter conviction is vulnerable to criticism. Perhaps proof, like Zen Buddhism (as described by Alan Watts) “can have no positive definition. It has to be suggested by saying what it is not, somewhat like a sculptor reveals an image by the act of re- moving pieces of stone from a block.” Unfortunately, this makes find- ing proofs something like “a game in which the rules [have] been partially concealed.” How do we cope with this situation? When a proof is proposed, it is read by other mathematicians. As more and more people study it, test it, and work with it, it gradually achieves acceptance. In summary, although we can’t say exactly what a proof is, it’s always possible to proof proofs, that is, to test them. To test your proof, let others read it. See whether they are convinced. Problems 8. Describe a typical equivalence class for the equivalence relations de- scribed in problems 1–5. (Example: For congruence for triangles (Problem 1), a typical equiva- lence class consists of all triangles of a particular shape and size.)
- 22. “master” — 2012/6/18 — 10:53 — page 7 — #17 i i i i i i i i 1.1. How to Build a Number System 7 9. Prove that all the elements of Sa are equivalent to each other. (Hint: Let b and c be elements of Sa. Then b a and c a. Use the laws of equivalence to draw the desired conclusion.) 10. Prove that if a b, then Sa D Sb. 11. Prove that every element a of S is a member of exactly one equivalence class. Building number systems with equivalence classes: a discussion Let’s start with one of the simplest number systems: the integers, Z, consist- ing of the whole numbers—positive, negative, and zero—i.e., Z D f0; ˙1; ˙2; : : :g. Besides numbers, Z comes with the two operations of addition and multiplication, so we can add and multiply integers—always obtaining, as it happens, another integer as a result. Addition is an invertible operation in Z; that is, given integers p and q, it is possible to un-add p to q. For example, un-adding 5 to 7 gives 2. (Check this: take 2, add back the 5, and note that we do get 7.) “Un-adding,” of course, is usually called “subtraction”. Un-adding 5 is adding 5 to 7, the number 5 being the additive inverse of 5. The point here is that un-adding is not a problem in the integers: 5 is an integer. It is inside Z with all the other integers. Every integer has an additive inverse in Z. Multiplication is not invertible. We cannot un-multiply 7 by 5 because there is no integer x such that multiplying it by 5 gives 7. In symbols, the equation 5x D 7 has no integer solution. We can fix this by building a new number system, the rational numbers Q, where un-multiplication (except by zero!) is possible. (For the sake of this discussion, please forget, for a page or two, that you know about the rational numbers already.) Towards the goal of building from the integers a number system in which un-multiplication is possible, let F be the set of all symbols of the form p=q where p and q are integers and q is not zero. Call the symbols p=q fractions. Think of p=q as representing the result of un-multiplying p by q (if only this were possible), but at the moment p=q is a pure symbol symbolizing nothing. These so-called fractions name the numbers in our new system, the ra- tional numbers. But the set of fractions is not the same as the set of rational numbers because many fractions name the same number. How do I know that many different fractions name the same number?
- 23. “master” — 2012/6/18 — 10:53 — page 8 — #18 i i i i i i i i 8 1. Axioms for the Reals Here is one argument that suggests this must be true. In Z, multipli- cation is commutative. In our new number system, let us agree that un- multiplication will be commutative, a natural property we might desire any operation possess. We stipulate that commutativity hold for fractions and hope that nothing bad happens, where ‘bad’ means a contradiction appears. It happens that un-multiplication is sometimes possible even within Z. If an integer p factors, say p D bc, then we can un-multiply p by b, getting c. For example, we can un-multiply 12 by 4 getting 3—all within the inte- gers. Ah, but suppose that p factors in two different ways: p D ad D bc. In this situation, if we un-multiply p by b we get c, and if we then un-multiply c by d, we get the fraction c=d. If un-multiplication is commutative, this must be the same as un-multiplying p by d first (getting a), and then un- multiplying a by b getting a=b. Therefore, if un-multiplicationis commuta- tive, and ad D bc, then the fractions a=b and c=d name the same rational number. Let us write a=b c=d in this situation. This is an equivalence relation for F (see problem 6), and the new number system we seek is the set of equivalence classes Q D F=. In summary, 3=6, 4=8 and 23=46, for example, are all names for the same rational number, namely 1=2 or one-half (to use its simplest fractional name). We are so used to this particular equivalence relation that we call the fractions 4=8 and 23=46 equal without thinking anything of it. They are not equal, however, at least not as elements of F. (Look at them: 4=8 and 23=46 are different!) They are equal in the set Q, though, since they belong to the same equivalence class. The view of equivalence relations and equivalence classes that emerges from this discussion is this: An equivalence relation on S supplies S with a new definition of equality, since equivalence relations have the same al- gebraic properties as equality. If we use the equivalence relation in place of equality, then the original set S functions as a set of names for a new set of objects: the set of equivalence classes: S=. An essential feature of our equivalence relation for fractions is that a=b and c=d can be tested for equivalence by a calculation (does ad D bc?) carried out entirely within Z without using any un-multiplications. Un- multiplication is still a fictitious operation since we have not completed construction of the new number system Q. What else is needed to complete the creation of Q? Quite a lot of work! First is the problem of definition: the operations of addition and multiplica- tion must be defined for rational numbers, and it must be proved that they satisfy the same basic properties in Q as in Z, for example the commutative
- 24. “master” — 2012/6/18 — 10:53 — page 9 — #19 i i i i i i i i 1.1. How to Build a Number System 9 and associative laws. Then there is the embedding problem: verifying that the old system Z is contained inside the new system Q. This means that inside Q a copy of the integers Z must be found that behaves, so far as ad- dition and multiplication are concerned, exactly as the original integers do. Finally, it must be shown that un-multiplication is possible in Q since this was the reason for the construction of Q in the first place. We will not tackle all this right now as we will have similar tasks to complete while constructing the real numbers and various alternative real number systems. (You are now allowed to remember that you already know about the rational numbers.) The problem of definition One unfinished element of the construction of the rational numbers is worth pursuing here: the problem of how to define operations on a set of equiva- lence classes. For the symbols a=b, addition is defined by a=b C c=d D .ad C bc/=.bd/: This is the well-known law for the addition of fractions based on a common denominator bd of the two fractions. It defines addition not for the rational numbers themselves, however, but only for their symbolic names from F. Can this symbolic addition be applied to the equivalence classes? It can, if it is well-defined, meaning that if a=b a0 =b0 ; and c=d c0 =d0 ; then a=b C c=d a0 =b0 C c0 =d0 : In other words, for an operation defined on a set of symbols to be well- defined on the equivalence classes, it must be proved that equivalent sym- bols combined with equivalent symbols give equivalent results. The issue of well-definedness arises whenever a mathematical system is defined us- ing equivalence classes.
- 25. “master” — 2012/6/18 — 10:53 — page 10 — #20 i i i i i i i i 10 1. Axioms for the Reals In praise of names It may seem that we belittle the fractions a=b by saying that they only are names for the rational numbers. If so, we should correct this impression right away. It is true that, for example, 20=25 and 80=100 and 580=725, and so on are names for an underlying ideal rational quantity (4=5), but for practical purposes we have to have these names in order to do anything with the rationals. In general, when a new number system S= is constructed using an equivalence relation, S is not discarded. We need those names. Problems 12. Prove that the symbolic addition of fractions is well-defined for the equivalence classes Q D F=. 13. For the symbolic fractions a=b of the set F, define multiplication by setting .a=b/.c=d/ D .ac/=.bd/: Prove that multiplication is well-defined. 14. The fractions in F of the form p=1 serve as embedded copies of the integers in the rationals. (a) Verify that equivalence on F for fractions of this form is the same as equality. (b) Verify that addition of fractions of this form is the same as the usual addition of integers. (c) Verify that multiplication of fractions of this form is the same as the usual multiplication of integers. Where do number systems really come from? The reader at this point may object that the process by which we have ob- tained the rational numbers from the integers cannot possibly be how the rational numbers were actually discovered. A few words about this now will place the theoretical description of number systems, to which this book is devoted, in the context of their historical development. The discovery and development of the rational numbers was a long pro- cess beginning before the dawn of written history. Because fractions are es- sential for commercial and astronomical calculation, they were discovered
- 26. “master” — 2012/6/18 — 10:53 — page 11 — #21 i i i i i i i i 1.1. How to Build a Number System 11 by many civilizations, including the ancient Egyptians and Babylonians. Over the centuries, rules for calculation with specific fractions slowly gave way to more general procedures. Many notations were invented and used. The current, standard notation was invented several times, most recently by Hindu mathematicians sometime before 600 AD. They used it without the bar separating numerator from denominator, which was added by Arab mathematicians later. In the history of the discovery of the rational numbers and other number systems, theory follows calculation. Thus, a theory of number was estab- lished, by the Greeks, only after centuries of systematic calculation. Eu- clid’s Elements (c. 300 BC) contains the first formal exposition of a theory of numbers (as well as a theory of geometry). One of the most important Greek discoveries about rational numbers, incidentally, is that there are ir- rational numbers. This is attributed to the Pythagorean school (c. 400 BC). After the Greeks, the theory of number systems is dominated by efforts to come to terms with various troublesome numbers: the negative, the irra- tional, and the complex—among others. In all this, the positive rationals remain uncontroversial. Although much was learned in the post-Grecian era about calculation with numbers, not much progress was made toward understanding their the- oretical nature until the nineteenth century. Then rapid progress was made as part of a general program undertaken by European mathematicians to place the ideas of the calculus on a firm foundation. All the classical num- ber systems received definitive treatment in the nineteenth century: the com- plexes first (c. 1800: Gauss and others), then the reals (1852–83: Dedekind and Cauchy), then the rationals (c. 1854–67:described by Bolzano and Han- kel as a system of numbers closed under addition, subtraction, multiplica- tion, and division), and, finally, the integers (1889: Peano). The equivalence relation on F appears first in an 1895 algebra text by Weber. The order in which the theory of these number systems developed is the reverse of the order in which they were discovered for computation and the reverse of the order in which they are usually studied. The theoretical development of number systems made possible the in- vention of many more systems in the 19th and the 20th centuries. In this book we describe five of the most important of these. They appear in histor- ical order: the complex numbers (1800 and earlier), the quaternions (1843), the constructive reals (1870–1930), the (1961), and the surreals (1970). Each of these systems was created in response to a computational challenge: the complexes in order to solve quadratic and higher degree equations, the
- 27. “master” — 2012/6/18 — 10:53 — page 12 — #22 i i i i i i i i 12 1. Axioms for the Reals quaternions in order to calculate with vectors and compute transformations of three-dimensional space, the constructive reals in order to deal with per- ceived limits on the nature of computation, the hyperreals in order to cal- culate with infinitesimals, and the surreals in order to evaluate positions in combinatorial games. Their discovery is a legacy of theoretical develop- ments of the 19th century including the description of the rationals as F=. For more on the history of number systems, see references [A1] and [A2]. For more on the history of mathematics in general, see [A3]. Summary Number systems have a long practical history but their theory developed only rather recently. One important theoretical development is the inven- tion of equivalence relations, which permit the elements of a set to serve as names for a new set constructed from the old. Equivalence relations ap- pear in many parts of mathematics. In this book they are used to create new number systems. 1.2 The Field Axioms The algebra of the reals In this and the next two sections we describe a categorical axiom system for the reals. The axioms of this system divide conveniently into three subsets: axioms for the algebra of the reals, axioms for the geometry of the reals, and a final axiom called completeness. Algebra is the most familiar of these; it is the natural place to start. Definition. Let S be a set with two operations called addition and mul- tiplication and written with the usual signs. We assume that the set S is closed under these operations, so that applying the operations to two ele- ments of S produces another element of S. Then S is a field if addition and multiplication have the properties: (a) For a, b, c in S, .aCb/Cc D aC.bCc/, — associativity of addition and .ab/c D a.bc/. — associativity of multiplication (b) For a, b in S,
- 28. “master” — 2012/6/18 — 10:53 — page 13 — #23 i i i i i i i i 1.2. The Field Axioms 13 a C b D b C a, — commutativity of addition and ab D ba. — commutativity of multiplication (c) For a, b in S, a.b C c/ D ab C ac. — distributivity (d) There are distinct, special elements 0 (zero) and 1 (one) in S such that for any b in S, 0 C b D b C 0 D b, — additive identity and 1b D b1 D b. — multiplicative identity (e) For any b in S, there is an element b in S so that b C . b/ D 0, — additive inverse and for any b in S, except 0, there is an element c in S so that bc D 1, — multiplicative inverse Properties (a)–(e) are the field axioms. They summarize the algebra of the real numbers, the rational numbers, and many other number systems. They were first collected together under the term ‘field’ by Weber in 1893. See reference [C7]. Assumption. Let Q be the set of all rational numbers, that is numbers of the form p=q, where p and q are integers and q is not zero. We assume that Q is a field. Problems 1. Let Z be the set of all integers. Z is not a field. List the axiom(s) not satisfied by Z. 2. Let n be a positive integer, and consider the set Z=n of integers modulo n, that is Z=n is the set of equivalence classes of Z under the equiva- lence relation x y .mod n/ if n divides x y with remainder zero. How many equivalence classes are there in Z=n? 3. Show that addition and multiplication are well-defined on Z=n, that is if a b .mod n/ and c d .mod n/, then a C c b C d .mod n/ and ac bd .mod n/.
- 29. “master” — 2012/6/18 — 10:53 — page 14 — #24 i i i i i i i i 14 1. Axioms for the Reals 4. Write out the complete addition and multiplication tables for Z=2, Z=3, Z=4, Z=5, and Z=6. 5. For what values of n is Z=n a field? (Hint: Problem 14 below describes the integral domain property, a property that all fields have. For which n does Z=n have the integral domain property.) 6. For rational numbers a and b define a b, if .a b/ is an integer. Is an equivalence relation? If so, are addition and multiplication well- defined? If so, is Q= a field? 7. Here is another relation on the rationals. Let a, b, c, d, be integers, with b and d not zero, and define a=b $ c=d if ad bc is even. Is $ an equivalence relation? If so, are addition and multiplication well-defined? If so, is Q=$ a field? 8. Let Q./ be the set of symbols of the form a C b, where a and b are rationals. (Call ‘blob’, then Q./ is ‘cue-blob’.) Define addition and multiplication in Q./ by setting .a C b/ C .c C d/ D .a C c/ C .b C d/; and .a C b/.c C d/ D .ac C 3bd/ C .bc C ad/ : Prove that Q./ is a field. (Hint: To find the multiplicative inverse of the element .a C b/ in Q./ solve the equation .a C b/.c C d/ D 1 C 0; for c and d by solving a pair of linear equations.) 9. Let Q.x/ be the set of all rational functions f .x/ D p.x/ q.x/ ; where p.x/ and q.x/ are polynomials with rational coefficients and q.x/ is not the zero polynomial. Let addition and multiplication in Q.x/ be the usual addition and multiplication of functions. Show that Q.x/ is a field.
- 30. “master” — 2012/6/18 — 10:53 — page 15 — #25 i i i i i i i i 1.2. The Field Axioms 15 Solving equations with the field axioms The field axioms justify many algebraic techniques for solving equations. The problems show how. Problems 10. Prove the cancellation law of addition: in any field S, if aCb D aCc, then b D c. (Hint: Use an inverse operation.) 11. Prove the cancellation law of multiplication: in any field S, if ab D ac and a ¤ 0, then b D c. 12. Explain how to solve the linear equation ax C b D c for x, where a, b and c are elements in a field S and a ¤ 0, using the cancellation laws of addition and multiplication. 13. Solve these linear equations: (a) 1 2 x C 1 3 D 4 9 , for x in the field Q, (b) 2 C 3 a C b C 1 4 D 3 C , for a C b in the field Q, (c) x2 1 x f .x/ C x3 2x C 3 x2 C 1 D x4 C 4x2 5 x2 C 1 ; for f .x/ in the field Q.x/. 14. Prove the integral domain property: in a field S, if ab D 0, then either a D 0 or b D 0. 15. Explain how to solve the quadratic equation x2 C .a C b/x C ab D 0, where a and b are elements in a field S and x is unknown, using the integral domain property. 16. Solve these quadratic equations: (a) x2 C 4x 60 D 0, for x in the field Q, (b) .a C b/2 C .2 C 2/.a C b/ C .3 C 2/ D 0, for .a C b/ in the field Q./, (c) f .x/2 x3 4x2 C x 2 x2 4 f .x/ x3 x2 C x x2 4 D 0, for f .x/ in the field Q.x/. (Hint: Factor the left-hand side of these equations in the usual way by factoring the constant terms, e.g., 60 in equation (a). Remember that 3 D 2 , in Q./.)
- 31. “master” — 2012/6/18 — 10:53 — page 16 — #26 i i i i i i i i 16 1. Axioms for the Reals How to find a proof There is no sure way to find proofs, but there are general principles, called heuristics, useful in all problem-solving situations. They in- clude: 1. Talk to yourself. Ask: What do I know already about this? Ask: Is there extra information I need and where can I get it? Ask: What if I could prove such-and-such, then can I prove what I want? 2. Work out a plan. Divide the proof into stages or cases that can be tackled separately. 3. Be flexible. Work both forward and backward, i.e., from the be- ginning or from the end. It may even be possible to start in the middle and work toward both ends. 4. Draw a picture. Also doodle, invent your own symbols, make up notation, build models. 5. Successively refine. Don’t expect your first thoughtsto be perfect in all detail. Scribble down ideas for later polishing. Go over the proof several times with a critical eye asking yourself: Is it convincing? 6. Indirect proof. If necessary, try proof by contradiction. The next exercises describe results that hold in all fields. Try heuristics on the proofs. Problems 17. Let n be a positive integer, and let a be an element of a field. Define na by setting na D a C a C a C C a: — exactly n a’s Prove that n.a C b/ D na C nb; .n C m/a D na C ma;
- 32. “master” — 2012/6/18 — 10:53 — page 17 — #27 i i i i i i i i 1.2. The Field Axioms 17 and .na/b D a.nb/ D n.ab/; where n and m are positive integers, and a and b are field elements. 18. Prove that the additive inverse of an element of a field is unique. 19. Prove that the multiplicative inverse of a non-zero element of a field is unique. 20. Prove that a D . 1/a and . a/ D a where a is an element of a field. (Hint: Use the uniqueness of additive inverses.) 21. For a nonzero element b in a field S, let b 1 stand for the multiplicative inverse of b. Prove that . b/ 1 D .b 1 /; .b 1 / 1 D b; and .ab/ 1 D b 1 a 1 : (Hint: Use the uniqueness of multiplicative inverses.) When are two fields the same? The answer is provided by the concept of isomorphism. Definition. Two fields S and T are (field) isomorphic if there is a one- to-one, onto function i WS ! T (i.e., a bijection) such that i.a C b/ D i.a/ C i.b/; and i.ab/ D i.a/i.b/: The function i is called a (field) isomorphism. Informally, the isomorphism of two fields, S and T , means that every algebraic operation in one of them (for example, a C b in S) is imitated by a parallel operation in the other (that is, i.a/ C i.b/ in T ).
- 33. “master” — 2012/6/18 — 10:53 — page 18 — #28 i i i i i i i i 18 1. Axioms for the Reals Problems 22. Show that the identity function i WS ! S is an isomorphism. 23. Show that the fields Z=p for different p are not isomorphic to each other. 24. Prove that field isomorphism is an equivalence relation on a set of fields. Summary The field axioms set forth algebraic laws that, as we shall see, are satisfied by the real numbers. We therefore expect them to be satisfied by any num- ber system that is an alternative to the reals. The field axioms are the basis for many of the usual algebraic techniques of solving equations. The con- cept of isomorphism makes precise when two fields have identical algebraic structure. 1.3 The Order Axioms The geometry of the reals Geometrically the reals are a straight line. This is expressed by order ax- ioms. Definition. A field S is (linearly) ordered if there is a subset SC of S satisfying: (a) If a and b are in SC , then so are a C b and ab, (b) If a is in S, then exactly one of the following is true: a D 0, a is in SC , or a is in SC (the trichotomy law). The elements of SC are called positive, while if x is in SC we call x negative. Two ordered fields S and T are (order) isomorphic if there is a field isomorphism i W S ! T such that a is in SC if and only if i.a/ is in T C . To say that the reals form a line means that given two real numbers, x and y, we can tell which is further along the line, that is, whether x y or y x. Thus, the linear nature of the reals takes the algebraic form of working with inequalities. Here is how inequalitiesare defined in an ordered field.
- 34. “master” — 2012/6/18 — 10:53 — page 19 — #29 i i i i i i i i 1.3. The Order Axioms 19 Definition. Let S be any ordered field. Let a and b be elements of S. We write a b; when b a is positive. We write a b when b a is positive or zero. Assumption. The rationals Q are an ordered field. The order axioms are not satisfied in many fields. Most of the fields we have introduced are not ordered. Problems 1. Prove that 1 is positive in an ordered field. 2. Show that the fields Z=p are not ordered. (Hint: 1 is positive by the previous exercise, so 1 C 1 is also positive. Argue that 0 is positive.) 3. Let .aCb/ be an element in the field Q./. Let us say that .aCb/ is a positive element of Q./ if either a and b are both positive, or a 0, b 0 and a2 3b2 , or a 0, b 0 and a2 3b2 . Show that Q./ is an ordered field. 4. For an element f .x/ D p.x/=q.x/ in the field Q.x/ of all rational functions the degree of f is the degree of p minus the degree of q. For example the degree of f .x/ D x2 1 x3 C x C 1 is 2 3 D 1. Show that defining the positive elements of Q.x/ to be those whose degree is positive does not make Q.x/ an ordered field. Solving inequalities with the order axioms The order axioms justify the usual algebraic laws of inequalities, including those in the next problems. Problems 5. Prove the trichotomy law for inequalities: given a and b in an ordered field S, exactly one of a D b, a b, and b a is true.
- 35. “master” — 2012/6/18 — 10:53 — page 20 — #30 i i i i i i i i 20 1. Axioms for the Reals 6. Prove the transitive law of inequalities: for a, b, and c in an ordered field S, if a b and b c, then a c. 7. Prove the law of addition of inequalities: in an ordered field if a b and c d, then a C c b C d. 8. Prove in any ordered field that if a b then b a. 9. In an ordered field, prove that the product of two negatives is positive, while the product of a negative and a positive is negative. 10. Let S be an ordered field. Prove that 1 C 1 ¤ 0. Deduce that between any two elements of S there are an infinite number of other elements. (Hint: Define 2 by 1 C 1. Given two field elements, a and b, show that their average .a C b/=2 lies between them.) Absolute value and distance Every ordered field has absolute values. They are used to define distance. Here is the definition. Definition. Let S be an ordered field and let a and b be elements of S. The absolute value is defined by jaj D ( a if a is positive or zero, a otherwise. The distance between a and b is defined as ja bj. The absolute value ja bj is the distance between a and b measured in units from the ordered field itself. The next few problems develop some properties of absolute values and distance. Problems 11. Prove that in any ordered field the absolute value function satisfies: ja C bj jaj C jbj; — triangle inequality and jnaj D njaj; where n is a positive integer.
- 36. “master” — 2012/6/18 — 10:53 — page 21 — #31 i i i i i i i i 1.3. The Order Axioms 21 12. Prove that in any ordered field: (a) The distance of any point to itself is the zero of the field. (b) The distance from a to b equals the distance from b to a. (c) Given three points, a, b, and c, the distance from a to c is less than or equal to the sum of the distance from a to b plus the distance from b to c. 13. In an ordered field, show that if jaj k, then k a k. Prove that if jb aj k, then a k b a C k. Embedding Theorems The next theorem is the first of many embedding theorems we prove. It describes how the natural numbers are embedded inside every ordered field. Theorem 1.3.1. Let S be an ordered field. For any natural number n define i.n/ D 1 C 1 C C 1: — exactly n ones where 1 is the multiplicative identity of S. Then i satisfies: (a) i.n C m/ D i.n/ C i.m/, (b) i.nm/ D i.n/i.m/, (c) i is one-to-one. Embedding the natural numbers in an ordered field S is executed by a function. The theorem says that every integer n has a copy of itself, i.n/, inside S. Parts (a) and (b) express the fact that given two natural numbers (n and m) their copies (that is, i.n/ and i.m/) in S behave arithmetically exactly like the originals. Part (c) says that the embedding is strict, i.e., one- to-one. As this is our first formal theorem, we give a complete proof, the heart of which is an argument by mathematical induction. Proof. The definition of i.n/ implies that i.n C 1/ D i.n/ C 1. This makes proof by induction work. Proof of (a). (By mathematical induction) Let s.n/ be the statement “i.n C m/ D i.n/ C i.m/ for all m.” Base Case: The statement s.1/ is “i.m C 1/ D i.m/ C 1”. This is clear from the definition of i.n/.
- 37. “master” — 2012/6/18 — 10:53 — page 22 — #32 i i i i i i i i 22 1. Axioms for the Reals Mathematical induction This powerful technique is used to prove a sequence of propositions, one for each natural number. Let s.n/ be some statement that depends on the value of the natural number n. Proof by induction consists of two steps: Base case: Prove s.1/. Inductive case: Assume that s.n/ has been proved. Use this as- sumption to prove s.n C 1/. What justifies this kind of proof? From a modern point-of-view, the validity of mathematical induction is a fundamental assumption concerning the nature of the natural numbers. Still, what do we to say to someone who challenges a proof by induction? It is not very satisfying simply to assert that we assume that this kind of proof is valid. If someone challenges proof by mathematical induction, ask them in return, “If you question my proof (that s.n/ is true for all n), then for what number n do you think s.n/ is false?” The rest of the con- versation might proceed along the following lines. “You say you doubt s.7/? But s.1/ is true, right, because I proved the base case. And you know that s.2/ follows from s.1/ because I proved the inductive case. Similarly s.3/ follows from s.2/, and s.4/ from s.3/, and so forth, and so on. Eventually, we reach 7. So s.7/ is true.” Eventually, we reach any particular natural number. This is one of Peano’s axioms for the natural numbers and is what ultimately justi- fies mathematical induction. Inductive Case: Assume that “i.nCm/ D i.n/Ci.m/ for all m” is true for a specific n. Then for n C 1 we have i .n C 1/ C m D i.n C m C 1/ D i.n C m/ C 1 — by definition of i.n C m/ D i.n/ C i.m/ C 1 — by assumption D i.n C 1/ C i.m/: — by definition of i.n/ Thus we have proved the statement s.n C 1/. This completes the proof of (a).
- 38. ‘‘master’’ --- 2012/6/18 --- 10:53 --- page 23 --- #33 i i i i i i i i 1.3. The Order Axioms 23 Proof of (b). (By mathematical induction) Let s.n/ be the statement “i.nm/ D i.n/i.m/ for all m.” Base Case: The statement s.1/ is “i.m/ D i.m/” which is clear. Inductive Case: Assume that “i.nm/ D i.n/i.m/ for all m” is true for a specific n. Then for n C 1 we have i..n C 1/m/ D i.nm C m/ — distributive law for integers D i.nm/ C i.m/ — by (a) D i.n/i.m/ C i.m/ — by assumption D .i.n/ C 1/i.m/ — distributive law in S D i.n C 1/i.m/: — definition of i.n/ Thus we have proved the statement s.n C 1/. This completes the proof of (b). Proof of (c). Suppose that i.n/ D i.m/. This means that 1 C 1 C C 1 (n times) D 1 C 1 C C 1: (m times) () As long as there is a 1 on each side of this equation we can cancel it from both sides. Eventually (after many cancellations, perhaps), one side or the other of the equation will be reduced to zero. If this happens on just one side, we get 1 C 1 C C 1 (q times) D 0: Because 1 is positive, the sum 1 C 1 C C 1 (q times) is also positive. By the trichotomy law for S, it is impossible that a positive number be zero. Therefore, as we cancel 1’s from both sides of (), both sides will become zero simultaneously, proving that the two sides of () have the same number of 1’s or that m D n. This proves that i is one-to-one. This embedding of the natural numbers in an ordered field can be ex- tended to the integers as described in the next theorem. Theorem 1.3.2. Let S be an ordered field. Let i be the embedding defined in Theorem 1.3.1. Extend i to negative integers and 0 by setting i.0/ D 0 and i. n/ D i.n/. Then i still has the properties (a) i.n C m/ D i.n/ C i.m/, (b) i.nm/ D i.n/i.m/, and (c) i is one-to-one.
- 39. “master” — 2012/6/18 — 10:53 — page 24 — #34 i i i i i i i i 24 1. Axioms for the Reals Problems 14. Where in the proof of the Theorem 1.3.1 did we use the fact that 1 is the multiplicative identity of S? 15. Prove Theorem 1.3.2. (Hint: The three parts of this theorem have already been proved for positive numbers. To prove them for negative numbers, survey all pos- sible cases. The result of Problem 13 in 1.2 may be useful. To prove (c) show that if i.n/ D i.m/, then i.n m/ D 0.) Summary The order axioms are the properties that characterize a straight line. They are the basis for the usual algebraic techniques for solving inequalities. Or- der isomorphism makes precise when two ordered fields have identical or- dered structure. An ordered field has an embedded copy of the integers. These copies of the integers combine according to the same laws of addition and multipli- cation as the integers themselves. From now on we always assume that an ordered field contains the integers and use n (rather than i.n/) to stand for the copy of n in that field. 1.4 The Completeness Axiom What makes the reals unique? The rationals Q and the reals R are ordered. So are many fields in between them. For example, the set of all numbers of the form a C b p 3, where a and b are rational numbers, is an ordered field that contains the rationals but does not contain all reals. This is the field Q./ of problem 8 in 1.2 and problem 3 of 1.3. What makes the real numbers unique among ordered fields is another property (or law or axiom) called completeness, which amounts to the assertion that the field has no gaps or holes. The purpose of this section is to make this precise and intelligible. There is no rational number in Q whose square is 2. This is what we mean by a hole: the rationals are missing the square root of 2, a number we can approximate to as many places as we like, that occupies, it seems,
- 40. “master” — 2012/6/18 — 10:53 — page 25 — #35 i i i i i i i i 1.4. The Completeness Axiom 25 –1 0 1 2 3 4 5 G upper bounds for G Figure 1.4.1. A set bounded above and some of its upper bounds. a definite place along a number line, but cannot be expressed as a fraction p=q where p and q are integers. Consider the set of rational numbers less than p 2. To be specific, let G D faja is rational and a2 2g: This set contains rationals whose square is arbitrarily close to 2, but no number whose square equals 2. (See Figure 1.4.1.) The set G is bounded above in the sense of the following definition. We use the bounds of G to get at the missing p 2. Definition. Let S be an ordered field. A subset G of S is bounded above if there is an element k of S such that a k for all a in G. Then k is called an upper bound for G. An element b of S is a least upper bound (lub for short) for a set G if b is an upper bound for G and b k for all upper bounds k of G. For example, G in Figure 1.4.1 is bounded above by k D 3 (see problem 2). Although G is bounded above, G does not have a least upper bound in the rationals. In the reals, however, G has a least upper bound, which, after we construct the reals, turns out to be p 2). Problems 1. Prove that there is no rational number p=q whose square is 2. (Hint: Assume that p=q is in lowest terms, that is, that p and q have no common factor. Use the fact that 2 is prime.) 2. Prove that the set G D faja is rational and a2 2g has a rational upper bound. (Hint: 100, for example, is an upper bound. Try a contrapositive proof.) 3. Let H be a subset of an ordered field S, and suppose that H has least upper bound b. For a positive integer n prove that
- 41. “master” — 2012/6/18 — 10:53 — page 26 — #36 i i i i i i i i 26 1. Axioms for the Reals (a) b C 1=n is not in H, but is an upper bound for H, (b) b 1=n is not an upper bound for H, (c) there is an element g of H such that b 1=n g b. 4. Verify that G (Figure 1.4.1) has no rational least upper bound with- out using the square root of 2, i.e., give a proof using only rational numbers. 5. Find an example of a set of rationals that has a rational least upper bound. 6. Define greatest lower bound (glb) and find an example of a set of ra- tionals that is bounded below but has no rational greatest lower bound. 7. Find and describe a set of rationals whose least upper bound is p 5. Do this without explicitly using the number p 5. 8. Find and describe a set of rationals whose least upper bound is e. Do this without explicitly using the number e. (Hint: Use an infinite series for e.) 9. Find and describe a set of rationals whose least upper bound is . Do this without explicitly using the number . Completeness The completeness axioms uses the terminology of boundedness and least upper bounds. Completeness Axiom. An ordered field S is called (order) complete if every non-empty subset of S with an upper bound has a least upper bound in S. If we accept that the least upper bound of G is p 2, then by insisting that the reals be complete we force p 2 to be a real number. Problems 10. Show that if a field S is complete, then a subset of S that is has a greatest lower bound. 11. In a complete, ordered field S, for each a in SC there is a unique integer n such that n a n C 1 called the greatest integer in a and denoted dae. Prove the existence and uniqueness of dae.
- 42. “master” — 2012/6/18 — 10:53 — page 27 — #37 i i i i i i i i 1.4. The Completeness Axiom 27 Order completeness and the Archimedean property To understand completeness, we introduce some related properties, the first of which is the Archimedean property. Suppose we have obtained a ruler marked in units of length a in order to measure a line of length b, where a and b are positive elements from an ordered field. (See Figure 1.4.2.) The Archimedean property asserts that we can measure the length b using a ruler marked in units of a provided the ruler is long enough. More precisely, the Archimedean property says that there is a natural number n such that if the ruler is at least n units of length a long, then it can measure any length up to b. Here is the formal statement. Definition. An ordered field S is Archimedean if, given two positive numbers a and b, there is a positive integer n such that b na. b a ruler marked in units of a Figure 1.4.2. Units, a ruler, and a length to be measured by it. It may seem obvious that a ruler marked in one length can measure any other length but this cannot be proved using the axioms of an ordered field alone. A further axiom or assumption is needed. The Archimedean property is linked to the existence (or not) of infinite elements. As it happens, the axioms for an ordered field don’t prevent it from containing infinitely large elements. (An infinitely large element is an element b of an ordered field S larger than all the natural numbers n1 embedded in S.) The hyperreals and the surreals, for example (see Chap- ters 6 and 7), contain infinitely large numbers. In an Archimedean field, however, this cannot happen. There, for any positive b, there is a positive integer n such that b n1. The Archimedean property is also linked with completeness: order com- pleteness implies that a field is Archimedean: Theorem 1.4.1. A complete field is Archimedean.
- 43. “master” — 2012/6/18 — 10:53 — page 28 — #38 i i i i i i i i 28 1. Axioms for the Reals Problems 12. In an Archimedean field S, prove that for a in SC there is an integer n such that 0 1=n a. This says that an Archimedean field does not contain elements that are infinitely small. 13. Prove Theorem 1.4.1. (Detailed hint: Let T D fn 2 S j n is a positive integer and na D bg. Show that T is bounded above by 1 C b=a. If T is empty there is nothing to prove (why?). Otherwise, T is non-empty and bounded above, so that by the completeness of S, T has a least upper bound, say M. Now M 1 is not an upper bound for T (why?), so there exists an integer k in T such that M 1 k M: The integer n D k C 1 is greater than M, hence not in T (why?). Therefore na b, which is the desired conclusion.) 14. Prove that the rationals are Archimedean. Order completeness and Cauchy completeness Completeness, as defined above, is called order completeness, because it is formulated in terms of the order of the field. Another kind of completeness, Cauchy completeness, is formulated in terms of distances and is related to the familiar notion of limit. Definition. An infinite sequence of numbers ˚ x.n/ from an ordered field S has limit b (or converges to b) if, given any element k of SC , there is an integer N such that jx.n/ bj k; for n N: The sequence ˚ x.n/ is a Cauchy sequence if, given any element k of SC there is an integer M such that jx.n/ x.m/j k; for m; n M:
- 44. “master” — 2012/6/18 — 10:53 — page 29 — #39 i i i i i i i i 1.4. The Completeness Axiom 29 The terms of a Cauchy sequence get closer and closer to each other as you move out in the sequence. In this way, the sequence is trying to converge, so to speak. The definition of Cauchy sequence, however, makes no mention of a limit, and, in fact, it is possible that a sequence be Cauchy but not have a limit in the field. This does not happen, however, in a Cauchy complete field according to the following definition. Definition. An ordered field S is Cauchy complete if every Cauchy se- quence in S has a limit in S. Cauchy completeness is similar to (order) completeness in intention. Both require that S be without holes of some sort. One difference between the two is that Cauchy completeness is defined using only the concept of distance which can be defined in some contexts without using order (for example in a plane). Cauchy completeness is sometimes called metric com- pleteness, to emphasize its connection with distance. Problems 15. Find an example of a sequence of numbers from an ordered field of your choice that is Cauchy but does not have a limit in that field. 16. If a sequence has a limit, prove that it is a Cauchy sequence. 17. Prove that a Cauchy sequence is bounded above and below. Order completeness implies Cauchy completeness: Theorem 1.4.2. A complete field is Cauchy complete. Lemma. In a complete field, every bounded monotonic (i.e., increasing or decreasing) sequence converges. Proof of the lemma. See problem 19. Proof of the theorem. Let S be a complete field. Let ˚ x.n/ be a Cauchy sequence in S. The goal is to prove that this sequence has a limit. By prob- lem 17, the set ˚ x.n/ is bounded above and below. Therefore, by (order) completeness, we can define y.1/ D lub ˚ x.1/; x.2/; x.3/; : : : ;
- 45. “master” — 2012/6/18 — 10:53 — page 30 — #40 i i i i i i i i 30 1. Axioms for the Reals and y.2/ D lub ˚ x.2/; x.3/; x.4/; : : : ; and, more generally, y.n/ D lub ˚ x.n/; x.n C 1/; x.n C 2/; : : : : Note that y.n/ is an upper bound for all the numbers for which y.n C 1/ is least upper bound, namely, y.n/ is the least upper bound of a set consisting of just one more number than the set of numbers of which y.n C 1/ is the least upper bound. Therefore y.n/ y.n C 1/: The new sequence ˚ y.n/ is nicer than ˚ x.n/ because ˚ y.n/ is mono- tonically decreasing and is bounded below by glb ˚ x.n/ . Therefore, by the lemma, ˚ y.n/ converges to an element b. We conclude the proof by showing that ˚ x.n/ also converges to b. Thus let k be an element of SC . Because ˚ x.n/ is Cauchy there is an integer N such that for n; m N , jx.m/ x.n/j k=2: In view of the definition of y.n/, this means that jy.n/ x.n/j k=2 for n N . (See problem 18). Next because ˚ y.n/ converges to b, we can assume that N is so large that for n N jy.n/ bj k=2: Now for n N we have jx.n/ bj jx.n/ y.n/j C jy.n/ bj k: This proves the theorem. Problems 18. Let G be a set of numbers bounded above. Let y D lub G. Prove that if jx C bj k for all x in G, then jy C bj k. (Hint: Rewrite the conclusion as two inequalities without the absolute value: k y C b k. Prove them separately.) 19. Prove the lemma. (Hint: Apply the completeness axiom to a bounded monotonic se- quence. Show that the sequence converges to its greatest lower bound, if the sequence is decreasing, or least upper bound, if it is increasing.)
- 46. “master” — 2012/6/18 — 10:53 — page 31 — #41 i i i i i i i i 1.4. The Completeness Axiom 31 [ , a1 [ , a2 [ , a3 b3] b4] [ , a4 S G b1] b2] Figure 1.4.3. Proof by bisection: the first few intervals. Completeness = Cauchy Completeness + the Archimedean property The Archimedean property and Cauchy completeness are consequences of order completeness. Conversely, together they imply order completeness. This is useful as it allows two simpler ideas to replace the difficult con- cept of order completeness. The proof of this theorem is also significant. It employs a powerful technique called proof by bisection. Theorem 1.4.3. An ordered field that is Archimedean and Cauchy complete is order complete. Proof. Let S be a linearly ordered field that is both Archimedean and Cauchy complete. Let G be a non-empty subset of S that is bounded above. Our goal is to prove that G has a least upper bound. The proof uses a trick; a trick so useful that it has become a technique: proof by bisection. We start with an interval Œa1; b1, which is then divided in half at its midpoint .a1 Cb1/=2. One of the resulting half intervals is cho- sen, call it Œa2; b2, and subdivided in turn into subintervals that are quarters of the original interval. One of these subintervals, Œa3; b3, is chosen and in turn divided in half and one of those halves chosen. Continuing to choose subintervals and halve them, the ith interval obtained is called Œai; bi . This is the setup for proof by bisection. It is illustrated in Figure 1.4.3. What we have not yet explained is how to pick the very first interval Œa1; b1 and how to choose between the two subintervals at each step be- fore proceeding to the next bisection. These choices depend on what one is trying to prove. However, in all proofs by bisection the left endpoints ai increase and the right endpoints bi decrease. In the middle is a point c that is the limit of both sequences of endpoints. Under favorable circumstances, c has whatever property is needed to complete the proof. For the proof of Theorem 1.4.3 we choose Œai ; bi so that bi is an upper bound for G and ai is not. (See Figure 1.4.3.) The proof is completed by
- 47. “master” — 2012/6/18 — 10:53 — page 32 — #42 i i i i i i i i 32 1. Axioms for the Reals proving: 1. it is possible to choose the subintervals Œai ; bi as described, 2. the sequence of upper endpoints fbi g is a Cauchy sequence (this uses the Archimedean property of S), 3. the sequence fbi g converges to a limit b (this uses the Cauchy com- pleteness of S), and, finally, 4. b is the least upper bound of G. Proof of (1). We are given that G is a non-empty subset of S that is bounded above. Let g be an element of G (G is non-empty), let a1 D .g 1/, and let b1 be an upper bound of G (G is bounded above) greater than a1. Then a1 is not an upper bound for G but b1 is, so Œa1; b1 can be our first interval. In order to choose the next subinterval, let d D .a1 Cb1/=2. Either d is an upper bound for G or not. If d is an upper bound, then set a2 D a1, and b2 D d; if d is not an upper bound, then set a2 D d and b2 D b1. In either case Œa2; b2 is a subinterval consisting of half of the first interval Œa1; b1 and such that the left endpoint a2 is not an upper bound for G, while the right endpoint b2 is an upper bound for G. The choices of Œa3; b3, Œa4; b4, and so on proceed in a similar manner. Proof of (2). Let L1 D b1 a1 be the length of the first interval. If Li is the length of the ith interval, then Li D L1=2i . Let k be an element of SC . To prove that fbig is a Cauchy sequence, we need to find an integer N such that jbi bj j k for all i; j N . Here it suffices to show that there is an integer N such that LN k since the terms of the sequence fbig are in the N th subinterval from i D N on. By the Archimedean property, there is an integer N such that L1=N k. Furthermore, 2N N (see problem 20), so that LN D L1=2N L1=N k: Proof of (3). Since fbig is Cauchy, it follows immediately from the Cauchy completeness of S, that the sequence converges to a limit c in S. Proof of (4). It is clear that c is also the limit of the sequence fai g. Therefore given a positive integer k there are elements an and bm such that c 1=k an c bm c C 1=k: () (See Figure 1.4.4.) The inequality () can then be used to show:
- 48. “master” — 2012/6/18 — 10:53 — page 33 — #43 i i i i i i i i 1.4. The Completeness Axiom 33 bm G an c 1/k S 1/k Figure 1.4.4. Proof by bisection at a later stage. (a) that every element g of G satisfies g c and (b) that every upper bound B of G satisfies c B. In other words c is an upper bound for G, and is the least upper bound. To prove (a) suppose, contrary to what we want to prove, that there is an element g of G such that g c. By the Archimedean property there is an integer k such that 1=k g c. Then, by (), there is an element bm such that c bm c C 1=k c C .g c/ D g; which contradicts the fact that bm is an upper bound for G! The proof of (b) is similar. (See Problem 21.) Corollary. A field is complete if and only if it is Cauchy complete and Archimedean. In other words, order completeness is equivalent to metric completeness plus the Archimedean property. Problems 20. Prove that 2N N for every natural number N . (Hint: Try proof by induction.) 21. Complete the proof of Theorem 1.4.3 part (4) (b). Summary Completeness is a technical condition whose purpose is to ensure that an ordered field have no gaps. It restricts the field in two ways that relate to properties of the field on a small scale. A complete ordered field cannot,
- 49. “master” — 2012/6/18 — 10:53 — page 34 — #44 i i i i i i i i 34 1. Axioms for the Reals for example, have an infinitely small element (the Archimedean property), and if the elements of a sequence from a complete ordered field are getting closer and closer to each other, then there is an element of the field toward which the sequence converges. Completeness completes the list of axioms for the reals. In the next chapter, we prove that the reals actually exist, by constructing them, and prove that our axiom system is categorical.
- 50. “master” — 2012/6/18 — 10:53 — page 35 — #45 i i i i i i i i 2 Construction of the Reals Why construct the reals? We will prove the existence of the real numbers two times—by twice con- structing a complete, linearly ordered field. Afterwards, we prove that all complete, linearly ordered fields are isomorphic, meaning that our axiom system for the reals is categorical. The reader might well object to this chapter in the following terms: “I’ve used the reals all my life; their properties are familiar; I know how to cal- culate with them; I know the calculus; I know everything. Why should I bother to construct the reals, when I know the result in advance?” How do we know the reals exist? Just because we’ve used a few of them over the years and assumed some properties for them does not allow us to conclude that they are really out there. Physicists argued for years about properties of the æther before it was discovered that it doesn’t exist. To avoid a similar fate the reals’ existence must be proved. The simplest and most reliable way to do this is to construct them. It was empirical evidence (the Michelson-Morley experiment, 1887) that did the æther in. The reader might argue that generations of mathemati- cians, scientists, and engineers have used the reals without encountering any problems: doesn’t that constitute experimental evidence for the existence of the reals? Certainly it does, but the goal of mathematics is to construct the- ories backed by stronger evidence than experiment. Experimental evidence only makes the existence of the reals plausible. It is possible, despite years of use, that the axioms of the reals contain a hidden inconsistency. By carry- 35
- 51. “master” — 2012/6/18 — 10:53 — page 36 — #46 i i i i i i i i 36 2. Construction of the Reals ing out a construction, we prove that the axioms of the reals are consistent, or, more precisely, are at least as consistent as the axioms of set theory. Furthermore, by constructing the reals we learn more about them and we learn methods that apply to the construction of other number systems. This is the primary reason for constructing the reals here. OK, but why construct the reals twice? Well, perhaps two constructions is overkill. Certainly, reading this book for the first time, one may skim over (or even entirely omit) one construction. However each construction furnishes ideas used later to construct alterna- tive number systems. One must read at least one of the constructions to understand why the axioms of the reals are categorical. That proof depends on a construction of the reals (though it doesn’t matter which). 2.1 Cantor’s Construction Real numbers are sequences of rationals Any construction of the reals from the rationals must fill in the gaps in the rational number system. One well-known way to get at these gaps is through approximation. For example , a fairly famous irrational, is approximated more and more closely by the familiar sequence of fractions: 3; 31 10 ; 314 100 ; 3141 1000 ; 31415 10000 ; 314159 100000 ; ; derived from its decimal expansion. In Cantor’s construction of the reals (first published in 1883) a sequence like this is treated as a single num- ber. Cantor’s idea is to define the reals as the set of all approximating sequences of rationals, and to manipulate sequences as though they were numbers. This means defining algebraic operations and a linear ordering for sequences. One difficulty the reader will spot right away is that many sequences ap- proximate the same number. For example also approaching is the rational sequence 3; 22 7 ; 333 106 ; 355 113 ; 103993 33102 ; ; derived from its continued fraction expansion. To deal with this, we impose an equivalence relation on the set of sequences. The equivalence classes actually form the real number system.
- 52. “master” — 2012/6/18 — 10:53 — page 37 — #47 i i i i i i i i 2.1. Cantor’s Construction 37 Two types of rational sequences are needed to carry out Cantor’s ideas in detail: Cauchy sequences (already introduced in Chapter 1), and null se- quences (defined below). Definition. A null sequence is a convergent sequence whose limit is zero. The next theorem gives the most important properties of these sequences. Theorem 2.1.1. Let x D fx.n/g and y D fy.n/g be sequences of rationals. 1) If x and y are Cauchy sequences, then so are fx.n/ C y.n/g and fx.n/y.n/g. 2) If x and y are null sequences, then so are fx.n/Cy.n/g and fx.n/y.n/g. 3) If x is a Cauchy sequence and y is a null sequence, then fx.n/y.n/g is a null sequence. Proof of (1): To prove that fx.n/ C y.n/g is a Cauchy sequence, let k be a positive rational number. We must find M so that j.x.n/ C y.n// .x.m/ C y.m//j k; for m; n M: In other words, we must make j.x.n/Cy.n// .x.m/Cy.m//j small (that is, less than k) as m and n get large (i.e., become greater than M). Well, using the triangle inequality for absolute values j.x.n/ C y.n// .x.m/ C y.m//j D jx.n/ x.m/ C y.n/ y.m/j jx.n/ x.m/j C jy.n/ y.m/j: The advantage of this is that if the two quantities jx.n/ x.m/j and jy.n/ y.m/j are separately made small (say less than k=2) then their sum will be small and, by the above inequality, j.x.n/ C y.n// .x.m/ C y.m//j will also be small. This is what we want. Now jx.n/ x.m/j and jy.n/ y.m/j can be made small separately because, by hypothesis, x and y are Cauchy sequences. Here is how the proof is completed: Given k; k=2 is also a positive ra- tional. Therefore, by the definition of Cauchy sequence, there are integers M1 and M2 such that jx.n/ x.m/j k=2 for m; n M1;
- 53. “master” — 2012/6/18 — 10:53 — page 38 — #48 i i i i i i i i 38 2. Construction of the Reals and jy.n/ y.m/j k=2 for m; n M2: Let M be the larger of the two integers M1 and M2. Then for m; n M, we have j.x.n/ C y.n// .x.m/ C y.m//j D jx.n/ x.m/ C y.n/ y.m/j jx.n/ x.m/j C jy.n/ y.m/j k 2 C k 2 D k: This proves that fx.n/ C y.n/g is a Cauchy sequence. For the product sequence xy D fx.n/y.n/g the trick of adding and subtracting the same term must be used before the triangle inequality can be applied: jx.n/y.n/ x.m/y.m/j D jx.n/y.n/ x.n/y.m/ C x.n/y.m/ x.m/y.m/j jx.n/y.n/ x.n/y.m/j C jx.n/y.m/ x.m/y.m/j D jx.n/jjy.n/ y.m/j C jx.n/ x.m/jjy.m/j: Thus we can make jx.n/y.n/ x.m/y.m/j small if we can make jx.n/ x.m/j and jy.n/ y.m/j both small (as before), and if jx.n/j and jy.m/j aren’t large. Now, in fact, jx.n/j and jy.m/j are bounded above and below (according to problem 11 in 1.4), so this strategy will succeed! Here is how the proof is completed: Let B be an integer bounding x and y, that is, such that jx.n/j B and jy.n/j B. Given the positive rational k; k=2B is also positive, so by the definition of Cauchy sequence there are integers M1 and M2 such that jx.n/ x.m/j k 2B for m; n M1 and jy.n/ y.m/j k 2B for m; n M2:
- 54. “master” — 2012/6/18 — 10:53 — page 39 — #49 i i i i i i i i 2.1. Cantor’s Construction 39 Let M be the larger of M1 and M2. Then for m; n M, we have jx.n/y.n/ x.m/y.m/j jx.n/j jy.n/ y.m/j C jx.n/ x.m/j jy.m/j B k 2B C k 2B B D k: This proves that fx.n/y.n/g is a Cauchy sequence. Problems 1) Prove Theorem 2.1.1 part (2). 2) Prove Theorem 2.1.1 part (3). Proof by trick? Proofs often use devices so surprising that they deserve to be called tricks. But a trick used a second time becomes a technique. Every branch, sub-branch and sub-sub-branch of mathematics has its characteristic tricks. Upon repeated use they become techniques. While working with the real numbers we have already seen two such tricks/techniques: adding and subtracting something and proof by bi- section. Equivalence of Cauchy sequences Here is the equivalence relation converting the set of Cauchy sequences into the real numbers. Definition. Two sequences x D fx.n/g and y D fy.n/g are equivalent, written x y, if the difference fx.n/ y.n/g is a null sequence. Theorem 2.1.2. Equivalence of sequences is an equivalence relation. This leads to Cantor’s definition of the real numbers: Definition. Let S be the set of all Cauchy sequences of rationals. The set of real numbers R is the set of equivalence classes, S.n/.
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- 56. Nach V e r e d a r i u s [30] repräsentierten 1884 die oberirdischen u n d versenkten Linien z u s a m m e n eine Länge von rund 1200000 km mit 3650000 km Leitungsdrähten. Letztere würden also hinreichen, n e u n z i g m a l den Gleichen zu umspannen. Der G e s a m t a u f w a n d a n K a p i t a l für Land- und Seetelegraphen betrug nach M u l h a l l bis zum Dezember 1882 rund 88 Mill. Pfd. St. = 1760 Mill. Mk. 4. Gebührentarif für Telegramme von Stationen des Deutschen Reiches[31] . (Für den billigsten und gebräuchlichsten Weg berechnet.) A. Die Wortlänge ist festgesetzt auf 15 Buchstaben oder 5 Ziffern im Verkehr mit: Wort- Taxe Mark. Deutschland (innerer Verkehr) 0,06 Afrika (West-): Kanarische Inseln 1,45 Senegal 2,65 Bolama 5,85 Bissao u. Konakry 5,90 Algerien-Tunis 0,27 Belgien 0,10 Bosnien-Herzegowina 0,20 Bulgarien 0,25 Dänemark 0,10 Frankreich 0,15 Gibraltar 0,25 Griechenland a) Festland und Insel Paros 0,40 b) nach den übrigen Inseln 0,45 Großbritannien und Irland 0,20 Außerdem ist für jedes Telegramm nach Großbritannien u. Irland eine Grundtaxe von 0,40 Mark zu erheben. Helgoland 0,15
- 57. Italien 0,20 Luxemburg 0,06 Malta 0,40 Montenegro 0,20 Niederlande 0,10 Norwegen 0,20 Österreich-Ungarn 0,10 Portugal 0,25 Rumänien 0,20 Rußland, europäisches u. kaukasisches 0,25 Schweden 0,20 Schweiz 0,10 Serbien 0,20 Spanien 0,25 Tripolis 1,05 Türkei 0,45 B. Die Wortlänge ist festgesetzt auf 10 Buchstaben oder 3 Ziffern im Verkehr mit: Wort- Taxe Mark. Afghanistan (via Bushire) 4,10 Afrika (Ost- und Süd-): Zanzibar 7,70 Mozambique u. Lorenzo-Marquez 8,75 Durban in Natal 8,70 den übrigen Anstalten Natals, Kapkolonie, Oranje-Freistaat 8,90 Transvaal 9,05 Ägypten: I. Zone Alexandrien 1,45 übrigen Anst. Nieder-Äg. 1,50 II. Zone (b. Wadi Halfa i. Nubien) 1,70 Suakim, via: Kabel Suez-Suakim 2,35 Annam (via: Bushire, Tavoy) 5,90 Arabien (Aden, Perim, Hedjas und Yemen) 3,60 Argentinische Republik (via: Lissab.) 7,25 Australien (via: Bushire, Penang): Süd-Australien 9,35 Victoria und Westaustralien 9,45
- 58. Neu-Süd-Wales 9,60 Queensland 9,85 Tasmania 10,05 Neu-Seeland 10,65 Balutschistan (via: Bushire) 4,10 Birma: Mandalay (via: Bushire) 4,50 Bolivien (via: Galveston): Cotagaita, Huanchaca, Potosi, Sucre (oder Chuquisaca), Tupiza 11,85 La Paz (über Mollendo) 13,55 Brasilien (via: Lissabon): nördl. Region: Pernambuco 7,25 Para 13,45 Fortaleza, Maranham u. den übrigen Anstalten 9,75 mittl. Reg.: (Bahia, Rio de Jan. etc.) 8,20 südl. Region: (Santos, Desterro, Rio Grande do Sul etc.) 8,95 Kapverdische Inseln: St. Vincent 4,00 Santiago 4,90 Chile (via: Lissabon): Valparaiso, Caldera, Concepcion, Copiapo, Coquimbo, Santiago, La Serena, Valdivia und Alt-Chile 9,15 Neu-Chile: Antofagasta u. Iquique 14,10 Arica und Tacna 15,65 China: Hongkong, Amoy, Foochow, Gutslaff, Saddle Island, Shanghai 7,00 Canton und Macao 7,45 Chinchow 7,90 Chining, Kiukiang und Puching 8,10 Chinkiang, Lingchow, Nanking, Nanning, Ningpo und Swatow 7,95 Chinkiangpoo, Lanchee, Nganking und Wuhu 8,00 Faltschan 7,65
- 59. Füng-Hwang-Ting 9,50 Hankow, Kinning und Tsinanfoo 8,15 Hweichow, Shaoking, Soochow und Woochow 7,85 Paoting-Foo 8,90 Newchwang 8,35 Ngouchow 7,50 Peking, Hoihow, Kiungchow, Tungschow und Yamchow 8,65 Tientsin, Chefoo, Kaiping, Liemschow, Pakhoi und Taku 8,25 Cochinchina, französisch (via: Bushire, Tavoy) 5,15 Columbien (via: Galveston): Buenaventura 4,90 den übrigen Anstalten 5,20 Corea (via: Rußl., Amur): Fusan 9,35 (via: Bushire, Tavoy): Binchong 9,00 Ichow 8,85 Jenchuan 9,30 Séoul (oder Han-Yang) 9,15 Costarica 4,60 Ecuador (via: Galveston) 8,05 Guatemala und Honduras 3,55 Guyana, Britisch (via Jamaica): Berbice 14,60 Demerara 14,50 Indien (via: Bushire): d. Anst. westl. v. Chittagong, ausschl. Ceylons 4,10 östl. v. Chittagong u. auf Ceylon 4,35 Isthmus v. Panama: Colon und Panama 4,40 Japan: Insel Tsushima 9,35 den übrigen Anstalten 7,70 Java und Sumatra (via: Bushire, Penang) 6,80
- 60. Madeira 1,60 Malacca (via: Bushire, Penang) 6,15 Mejico: Goatzacoalcos 2,75 Malamoras 0,80 Mejico City, Tampico u. Veracruz 1,75 den Anstalten der mejicanischen Bundesregierung 2,05 den Anstalten der Einzelstaaten und Privatgesellschaften 2,55 Nicaragua: San Juan del Sur 4,40 den übrigen Anstalten 4,60 Paraguay (via: Lissabon) 7,25 Penang (via: Bushire) 5,55 Persien, ausschließlich der Anstalten am Persischen Golf 1,30 Persischer Golf (via: Persien, Bushire): Bushire 2,45 den übrigen Anstalten 3,65 Peru (via: Galveston): Callao, Lima 7,55 Mollendo 10,65 Payta 8,35 Piura 8,50 Chancay, Chicla, Chosica, Huacho, Malucana, San Bartolome, San Mateo, Sta. Clara, Supe, Sucro 8,05 den übrigen Anstalten 11,60 Philippinen-Inseln: Luzon 8,85 Rußland, asiatisches: I. Region, westlich vom Meridian von Werkhne-Udinsk 1,45 II. Region, östlich von demselben 2,35 Bokhara 1,70 Salvador: Libertad 3,40 den übrigen Anstalten 3,55 Siam (via: Bushire, Tavoy) 4,75 Singapore (via: Bushire, Tavoy) 6,40
- 61. Tonking (via: Bushire, Tavoy) 6,30 Uruguay (via: Lissabon) 9,50 Venezuela (via: Galveston) 5,20 Vereinigte Staaten von Amerika, Britisch Amerika und St. Pierre- Miquelon: 1. Alabama, Arkansas, Canada (Ost- und West-), Cape Breton, Carolina (North- und South-), Columbia (District of), Connecticut, Delaware, Florida (und zwar: Jacksonville und Pensacola), Georgia, Illinois, Indiana, Iowa, Kentucky, Louisiana, Maine, Maryland, Massachusetts, Michigan, Minnesota, Mississippi, Missouri, New-Brunswick, Newfoundland, New-Hampshire, New-Jersey, New-York (einschließl. Stadt New-York), Nova Scotia, Ohio, Pennsylvania, Prince Edwards Island, Rhode Island, St. Pierre-Miquelon, Tennessee, Texas, Vermont, Virginia (Ost-), West-Virginia, Wisconsin 0,65 2. Arizona, California, Colorado, Columbia Britisch, Dakotah, Florida (ausgenommen Jacksonville u. Pensacola), Idaho, Indian Territory, Kansas, Manitoba, Montana, Nebraska, Nevada, New-Mejico, North- Western Territory, Oregon, Utah, Vancouver Island, 1,05
- 62. Washington Territory, Wyoming Westindien: Antigua 10,50 Barbados 12,35 Cuba, und zwar: Havanna 2,45 Cienfuegos 3,20 Santiago de Cuba 3,65 Bayama, Guantanamo und Manzanillo 3,90 den übrigen Anstalten 2,70 Dominica (Kleine Antillen-Insel) 11,05 Grenada 12,30 Guadeloupe 10,90 Jamaica 6,15 Martinique 11,35 Portorico 9,35 St. Croix 9,70 St. Kitts (St. Christoph) 10,25 St. Lucia 11,60 St. Thomas 9,45 St. Vincent, Westindien 11,85 Trinidad 12,80
- 63. Anhang. Das Fernsprechwesen[32]. 1. G e s c h i c h t e . Die Versuche, den Schall mittels geeigneter Übertragung der Schallwellen fortzuleiten, gehören schon einer ziemlich weit zurückliegenden Vergangenheit an. So weist der englische Elektriker P r e e c e nach, daß sein Landsmann, der Physiker R o b e r t H o o k e , bereits 1667 derartige, wenn auch noch ziemlich rohe Versuche anstellte, indem derselbe einen ausgespannten Faden benutzte. Einen telephonischen Apparat konstruierte auch W h e a t s t o n e im Jahre 1819. Aber erst 1861 fertigte der 1874 verstorbene Lehrer P h i l i p p R e i s in Friedrichsdorf bei Frankfurt a. M. das e r s t e e l e k t r i s c h e Te l e p h o n . Dieser von Reis mit dem von ihm selbst erfundenen Worte „Te l e p h o n “[33] bezeichnete Apparat übertrug musikalische Töne und Melodieen, ferner auch Worte, wenn schon in etwas unvollkommener Weise, auf ziemlich weite Entfernungen. Die ganze Sache wurde indes von den Physikern nur als eine Kuriosität, nicht als praktisch wichtig betrachtet[34], und auch Reis selbst hatte seinen Apparat von Anfang an nur für Unterrichtszwecke bestimmt. So kam es, daß der deutsche Erfinder und sein Instrument in Europa nach kurzer Zeit wieder vergessen wurden. In Amerika dagegen wurde der deutsche Gedanke weiter verfolgt. 1868 konstruierte dort ein gewisser v a n d e r W e y d e ein verbessertes Reissches Telephon, das deutlich, wenn auch nur schwach und mit näselndem Tone, hineingesprochene Worte übertragen haben soll. V a n d e r W e y d e setzte seine Versuche fort, und seinen Bestrebungen schloß sich E l i s h a G r a y in Chicago an. Aber all diese Telephone, wie auch die in England gefertigten, eigneten sich in der Hauptsache nur
- 64. zur Übertragung musikalischer Töne, nicht aber für artikulierten Schall, d. i. für die Wiedergabe der Sprache. Dieses so schwierige Problem wurde durch den Taubstummenlehrer G r a h a m B e l l in B o s t o n , einen geborenen Schotten, im Jahre 1876 glücklich gelöst und so die Welt von Amerika her mit dem p r a k t i s c h e n Telephon beschenkt. Seitdem gelang es, durch verschiedene Verbesserungen die telephonische Wirkung bedeutend zu erhöhen und überhaupt den Fernsprech-Apparat für die Verwendung im Verkehre noch bequemer zu gestalten. Großartiges zeigte bezüglich des Fernsprechwesens besonders die internationale elektrische Ausstellung zu Philadelphia im Jahre 1884. Der dort ausgestellte Quadruplex-Translator E d i s o n s z. B. verstärkte den Ton vierfach; sein Mikrophon[35] ließ den Schritt einer Fliege deutlich hören; das größte Aufsehen aber erregte unter den Laien sein lautsprechendes Telephon, dessen Töne im Umkreis von 30 Fuß deutlich vernehmbar waren, und dessen hohe Noten bedeutend ausgeprägter waren als die tieferen[36]. Sicher wird auch d i e Zeit nicht ausbleiben, wo man, wie schon R e i s andeutete, die menschliche Stimme übers Meer senden wird, wie das mittels des Telegraphen bezüglich der Schrift bereits der Fall ist[37].
- 65. Fig. 15. Philipp Reis. In Deutschland wurde das erste Fernsprechamt für den öffentlichen Verkehr am 12. November 1877 in Friedrichsberg bei Berlin eröffnet, und heute (Ende 1885) giebt es, dank der Thatkraft des obersten Leiters der deutschen Reichspost- und Telegraphenverwaltung, D r. v o n S t e p h a n s , in 81 Orten 12655 Fernsprechstellen und 21357 km Drahtleitungen[38]. Auch in den übrigen Kulturländern der Erde hat das Fernsprechwesen fast überall Eingang gefunden; selbst das Reich der Mitte hat sich nicht ausschließen können. Shanghai zählt bereits 77,
- 66. Hongkong 40 Fernsprechstellen; ja sogar die Hauptstadt der Sandwich-Inseln, Honolulu, hat ihre Telephonleitung[39]. 2. R e c h t s v e r h ä l t n i s s e . Die Rechtsverhältnisse im Fernsprechbetrieb sind sehr verschiedenartig. Ganz frei in Anwendung und Ausbeutung ist der Betrieb in den Vereinigten Staaten, in Schweden, Norwegen und in den meisten Kolonieen; g a n z v o m S t a a t a b h ä n g i g i m D e u t s c h e n R e i c h u n d i n d e r S c h w e i z ; unter der Kontrolle der Regierung in England, Rußland, Österreich, Frankreich, Italien, Spanien und Portugal. Doch ist in England, Rußland und Österreich der Betrieb den Privatgesellschaften auf l a n g e Zeit vertragsmäßig gewährt, in den anderen nur auf k u r z e Fristen. 3. S t a t i s t i s c h e s . Ein annäherndes Bild von der Verbreitung des Fernsprechers in der Mehrzahl der europäischen Staaten im Jahr 1885 giebt folgende Tabelle[40]: Länder. Städte mit Fernsprech- Einrichtungen. Abgerundete Zahl der Stellen. Jährlicher Abonnements- betrag in Mark. Deutschland 81 13000 150 England 180 (?) 12000 100–400 Frankreich etwa 20 10000 480 Italien 18 7000 92–140 Schweden 51 10000 128–216 Schweiz 30 5000 120–200 Spanien unbekannt 1000 80–200 Niederlande etwa 11 4000 136–204 Belgien 12 5000 160–200 Rußland 7 3000 560 Österreich-Ungarn 10 4500 180–300 In den V e r e i n i g t e n S t a a t e n von Amerika betrug 1884 die Länge der Telephonlinien 193120 km (vgl. Gothaischer Kalender für 1886). Im gleichen Jahre zählte N e w -Yo r k mit Umgebung schon 10600 Abonnenten, während gleichzeitig ganz England nur 11000–
- 67. 12000 aufwies, so daß demnach eine e i n z i g e Stadt Amerikas fast ebensoviel Telephon-Abonnenten besitzt, als ein ganzes Königreich in Europa. Die Z a h l d e r Te l e p h o n g e s e l l s c h a f t e n betrug in der Union im Jahre 1884 über 45, deren Anlagekapital 266,7 Millionen Franken (Journal télégraphique, 1885, S. 190–192). Die l ä n g s t e der Fernsprechanlagen in D e u t s c h l a n d ist zur Zeit diejenige zwischen Berlin und Hannover mit 341 km. In A m e r i k a dagegen wird bereits zwischen New-York und Chicago, d. i. auf eine Entfernung von 1600 km, mittels des Telephons korrespondiert. 4. B e d e u t u n g d e s F e r n s p r e c h e r s . In den wenigen Jahren, welche seit der Erfindung des einfachen und doch so wunderbar wirkenden Apparates verflossen sind, hat derselbe bereits eine Bedeutung erlangt, wie sie wohl keinem Verkehrsmittel der neuern Zeit in so kurzem Zeitraum zugemessen war. Die Telephone und Mikrophone haben nicht nur für den allgemeinen Verkehr der Bewohner großer Städte untereinander hervorragenden Wert, sondern ihre Anwendung erweist sich auch in vielen anderen Fällen als äußerst nutzbringend. G e s c h ä f t s h ä u s e r bedienen sich des Fernsprechers zur Vereinfachung ihres Geschäftsbetriebs. Höchst wichtige Dienste leistet er der P o l i z e i . Desgleichen eignet er sich vielfach für m i l i t ä r i s c h e Z w e c k e , so z. B. im Vorpostendienste, zur Verbindung eines „ballon captif“ mit der Erde. Auch im E i s e n b a h n d i e n s t e findet er mannigfache Verwendung. Für den Ta u c h e r wieder bildet das Telephon ein sehr bequemes Verständigungsmittel im Verkehre mit Personen zu Lande oder zu Schiffe. Ebenso spielt es schon im B e r g - und H ü t t e n w e s e n eine bedeutende Rolle. Seine große Empfindlichkeit führte ferner zur Verwendung desselben für ä r z t l i c h e Zwecke, und auch die W i s s e n s c h a f t wurde durch dessen Erfindung zu einer Reihe sehr interessanter Untersuchungen veranlaßt.
- 69. II. Weltpost. Erstes Kapitel. Geschichte des Postwesens[41]. I. Altertum. Die Staaten als solche, d. h. die Regierungen, hatten schon frühzeitig für ihre Zwecke bestimmte Anstalten zur Herstellung gesicherter und schneller Verbindungen errichtet. Dabei wurden anfänglich die im Dienste des Herrschers stehenden Boten von der Hauptstadt aus mit den Befehlen an die obersten Verwaltungschefs, die Truppenbefehlshaber u. s. w. in den Provinzen d i r e k t abgesandt, und sie brachten die Berichte auch wieder zurück. Sehr bald aber kam man auf den Gedanken der Errichtung von Stationen und des stationsweisen Transportes mittels Wechsels des Beförderungsmittels, wodurch zugleich eine erhebliche Beschleunigung erzielt wurde. Solche B o t e n a n s t a l t e n besaßen bereits die Regierungen in I n d i e n , C h i n a , Ä g y p t e n , A s s y r i e n , B a b y l o n i e n und die K ö n i g e d e r H e b r ä e r . In I n d i e n waren an den Endpunkten der ziemlich kurzen Stationen Hütten errichtet. Sobald ein Bote bei einer solchen Hütte ankam, empfing der schon bereitstehende andere das Schreiben, um
- 70. damit bis zur folgenden Station zu laufen. Jeder war mit einer Schelle versehen, auf deren Laut alle Begegnenden ausweichen mußten; zugleich kündigte damit der Bote seine Ankunft auf der Station an. Bei wichtigeren Depeschen oder gefährlichen Passagen gingen zwei Boten zur Erhöhung der Sicherheit. Zum Übersetzen über Gewässer bedienten sie sich, wo keine Brücken oder Fähren vorhanden waren, eines Schwimmgürtels. Alle zehn Stadien (¼ geogr. Meile) war auch eine Säule gesetzt, welche die etwaigen Nebenwege, sowie die Entfernungen anzeigte. Besondere Beamte standen überdies dem Verkehrs- und Straßenwesen vor. Von Ä g y p t e n erzählen die alten Geschichtschreiber, daß nach Vorschrift des Gesetzes jeder König früh aufgestanden sei und zuerst die eingegangenen Briefe gelesen habe. Bei den A s s y r e r n wird schon gelegentlich der Erzählung der Vorbereitungen, welche die Königin Semiramis zu ihrem großen Zuge nach Indien traf, der Boten gedacht, welche deren Briefe und Befehle beförderten. In Bezug auf B a b y l o n i e n heißt es im Alten Testament: „Nebukadnezar sandte von Ninive Botschaften zu allen, die da wohnten in Cilicien, Damaskus und auf dem Libanon, Karmel und in Kedar; auch zu denen in Galiläa und auf dem großen Felde Esdrelom; und zu allen, die da waren in Samaria, und jenseits des Jordan bis gen Jerusalem; auch in das ganze Land Gesem bis an das Gebirge des Mohrenlandes.“ Bei den H e b r ä e r n wurden während der Regierung der Könige die Schreiben derselben und die Berichte der Obersten und Ältesten ebenfalls durch besoldete königliche Boten befördert, die der Leibwache zugeteilt waren. „Und die Läufer gingen hin mit den Briefen von der Hand des Königs und seiner Obersten durch Israel und Juda“ (Hiskia, 728–699 v. Chr.). Ja sogar aus dem 10. Jahrhundert v. Chr. besitzen wir eine desfallsige Nachricht im ersten Buch der Könige: „Und sie (die Königin Isebel, Gemahlin Ahabs, 918–890 v. Chr.) schrieb Briefe unter Ahabs Namen und versiegelte
- 71. sie unter seinem Petschier und sandte sie zu den Ältesten und Obersten.“ Den nächsten Fortschritt nach der Zerlegung in Stationen bildete die A n w e n d u n g d e s P f e r d e s für den Kurierdienst. Die erste desfallsige Einrichtung ging der gewöhnlichen Annahme nach von dem P e r s e r k ö n i g e C y r u s aus und bestand hauptsächlich in folgendem: in einer Entfernung von ca. 4 zu 4 Parasangen (3–4 Meilen) waren Pferde und Reiter stationiert, von welch letzteren stets einer bereit zu sein hatte, um nach Einlauf eines königlichen Schreibens dasselbe in der schnellsten Gangart des Pferdes bei Tage oder bei Nacht, in der größten Hitze des Sommers oder im Schnee des Winters zur nächsten Station zu befördern. Außerdem war bei jeder Station ein Aufseher bestellt, dessen Aufgabe es war, die Briefe in Empfang zu nehmen, wieder zu übergeben, die ermüdeten Pferde und Männer zu beherbergen und frische abzusenden. Bei den Griechen sagte man, die persischen Postreiter flögen schneller als Kraniche, und Herodot versichert, daß nichts in dieser Welt geschwinder sei, als diese Reiter. Briefe konnten durch sie auf der großen Straße von Sardes nach Susa, die 450 Parasangen (337 Meilen) maß, welche wieder in 111 Stationen geteilt waren, in 5–7 Tagen befördert werden. Ein Fußgänger hingegen, der fünf Parasangen (3¾ Meilen) täglich zurücklegte, brauchte hierzu 90 Tage. Die gesamte Posteinrichtung nannte man angara, ein Wort, das soviel bedeutet als F r o n d i e n s t . Die Griechen entlehnten diese Bezeichnung von den Persern und überlieferten dieselbe ihrerseits wiederum an die Römer, so daß noch bis ins Mittelalter das Kurierwesen im Lateinischen mit angaria bezeichnet wurde. Der Chef der ganzen Anstalt war ein hoher, dem königlichen Hofe nahestehender Beamter. Darius Kodomannus, Persiens letzter König, bekleidete jenes hohe Amt vor seiner Thronbesteigung. Das Volk war von der Benützung der Anstalt ausgeschlossen; sie trug rein staatlichen Charakter; nur der König bediente sich ihrer zu seinen Regierungszwecken.
- 72. Wohl ebenso frühzeitig als in Persien, vielleicht noch früher, scheint die Verwendung des Pferdes zum Postdienste auch in C h i n a stattgefunden zu haben. Hierauf läßt besonders die schon in alten Zeiten sehr vorgeschrittene Organisation der Verwaltung des weitläufigen Reiches und das Vorhandensein trefflich angelegter und gut unterhaltener Straßen schließen. Frühzeitig schon wurde den G r i e c h e n die Buchstabenschrift und das Briefschreiben von Asien aus überliefert, aber die Einrichtung einer bestimmten Staatsverkehrsanstalt haben sie den asiatischen Monarchieen nicht nachgeahmt. Zunächst war Griechenland nicht ausgedehnt genug, um unter den damaligen Verhältnissen die Notwendigkeit einer solchen Anstalt empfinden zu lassen. Dann waren auch die durch die vielfachen Wanderungen der griechischen Volksstämme hervorgerufenen Erschütterungen der Entwicklung einheitlicher Institutionen hinderlich. Später kamen die häufigen Fehden und unerquicklichen Nergeleien der kleinen Republiken, der peloponnesische Krieg u. s. w., bis endlich die Schlacht von Chäronea (338 v. Chr.) der griechischen Unabhängigkeit ein Ende machte. Überdies ersetzte vielfach die sehr rege Schiffahrt die Landkommunikationen, wie das noch heute z. B. in Dalmatien, Norwegen, Chile, dem Sunda-Archipel u. s. w. der Fall ist. Endlich führte auch der allen Stämmen und Landschaften gemeinschaftliche religiöse Kultus gelegentlich der fast alljährlich stattfindenden Spiele und Nationalfeste Leute aus allen Gegenden, wo nur immer die griechische Zunge ertönte, zusammen und bot reichliche Gelegenheit dar, im gegenseitigen Verkehre die Gedanken auszutauschen und sich über die verschiedensten Verhältnisse mündliche Mitteilung zu machen. Infolge davon beschränkte sich die ganze Posteinrichtung des Landes auf die sogen. H e m e r o d r o m e n (= Tagläufer, vom griech. heméra = Tag, und griech. dremo, ich laufe) oder Schnellläufer, die nur aus besonderer Veranlassung abgesandt wurden, und deren sich nicht nur die Obrigkeiten, sondern auch Private bedienten. Diese Hemerodromen waren mitunter von erstaunlicher Geschwindigkeit, und die alten Schriftsteller erwähnen einzelner bei Namen. Phidippides, ein
- 73. Fig. 16. Hemerodrom. (Nach dem „Poststammbuch“.) Botenläufer von Gewerbe, sagt Herodot, legte den Weg von Athen nach Lacedämon (1200 Stadien = 30 geogr. Meilen) in zwei Tagen zurück. Nach der Schlacht von Salamis wurde der Platäer Euchidas nach Delphi gesandt, um, da das heilige Feuer erloschen war, reines Feuer zu holen. Die Entfernung hin und zurück beträgt 1000 Stadien (= 25 geogr. Meilen); er brauchte nur e i n e n Tag, starb aber infolge der Überanstrengung. Von Ladas, einem vielgenannten Läufer Alexanders von Macedonien, sagte man, daß seine Spuren im Sande kaum wahrnehmbar gewesen seien. Die Ausrüstung dieser Schnellläufer bildeten Bogen, Pfeile, Wurfspieß und Feuersteine. A l e x a n d e r d e r G r o ß e hatte bei dem Charakter seiner Regierung nur wenig für die Verkehrseinrichtungen zu thun vermocht. Als er die Hand an das Werk der innern Ordnung legen wollte, überraschte ihn der Tod. In den eroberten Ländern waren die früheren persischen Anstalten im allgemeinen in Wirksamkeit geblieben. Die R ö m e r waren ein eroberndes Volk; jede Nation, die sie sich unterwarfen, mußten sie daher wenigstens anfänglich durch die Gewalt der Waffen niederhalten. Um aber über ihre Legionen und Kohorten rasch verfügen, um sie schnell dorthin werfen zu können, wo der Staat sie nötig hatte, bedurften die Römer eines gut ausgebildeten und weitverzweigten Straßennetzes. In der That galt denn auch ein Land ihnen nur dann für vollkommen erobert, wenn es von Militärstraßen durchzogen war. Schon in den ersten Zeiten der Republik wurden deshalb alle Städte Latiums, sobald sie unter Roms Herrschaft gekommen, dann die Gebiete Campaniens, endlich die Bergstädte der besiegten Samniter durch vorzügliche Kunststraßen mit Rom verbunden. In erster Reihe waren nun diese
- 74. Straßen freilich nur für militärische Zwecke bestimmt; aber sie dienten doch schon frühzeitig auch dem V e r k e h r e . So gingen vor allem, wie in den übrigen älteren Reichen, s t a a t l i c h e B o t e n von Rom zu den auswärts bestellten Beamten und Befehlshabern, um Befehle oder Nachrichten zu überbringen, oder es wurden von diesen solche nach Rom gesendet. Die Boten hießen viatores, cursores, statores, tabellarii (letzterer Name rührt davon her, daß die Alten statt der Briefbogen Täfelchen [tabellae] benutzten). Die Vergütung, welche sie für die Übermittlung von Nachrichten erhielten, nannte man calcearium, Schuhgeld[42]. Eine bedeutende Förderung wurde dem Nachrichten- und auch Frachtenverkehr zu teil durch jene große Gesellschaft römischer Ritter, welche in den letzten Zeiten der Republik die Staatsländereien in den Provinzen, sowie die Zehnten, Gefälle und Steuern pachtete und einen ausgedehnten, schwunghaften Handel mit Getreide und anderen Landesprodukten betrieb. Diese Genossenschaft hatte ihren Centralsitz in Rom und ihre Niederlagen und Comptoire in allen wichtigeren Provinzstädten. Ihr Nachrichten- und Geldverkehr vom Mittelpunkte nach den Filialen und zwischen diesen selber wieder war ein großartiger, und deshalb unterhielt die Gesellschaft eine große Zahl von Briefträgern (tabellarii), welche Briefe und leichtes Gepäck bis in die kleinsten Städte aller Provinzen mit großer Schnelligkeit und ziemlicher Regelmäßigkeit beförderten. Diese Briefträger durften auch Sendungen von Privaten übernehmen und wurden häufig hierzu benützt. Außerdem gab es noch zahlreiche Privatboten. Reiche Familien, die in Rom wohnten, hatten große Güter in den Provinzen, oder ihre Söhne studierten an griechischen Schulen. Da sie nun mit ihren Verwaltern und ihren Kindern in regelmäßigem Verkehre bleiben wollten, so unterhielten sie Briefboten, die nicht bloß von ihnen, sondern auch von Bekannten mit Sendungen betraut wurden. Häufig wurden auch Reisenden, Schiffern, Kaufleuten, Fuhrleuten u. s. w. Briefschaften zur Abgabe in den Orten, wohin ihr Geschäft sie führte, übergeben. Freilich war diese Art der Beförderung in
- 75. hohem Grade unvollkommen. Wir ersehen das besonders aus den Briefen Ciceros an Atticus. Monatelang erhielt jener keinen der ihm vom Freunde geschriebenen Briefe, dann häufig drei oder vier auf einmal; nicht selten sind einige unterwegs abhanden gekommen; andere werden ihm eröffnet überbracht; später geschriebene erhält er eher als solche von früherem Datum; öfters ist er genötigt, mehrere Briefe des Atticus, die ihm in einem Zeitraum von vier bis fünf Monaten zugegangen waren, auf einmal zu beantworten, weil er keinen zuverlässigen Überbringer auffinden konnte. Alle diese Umstände führt Cicero in seinen Briefen immer nur nebenher und in dem Tone an, in welchem man von Dingen spricht, die sich ganz von selbst verstehen und alle Tage sich zutragen. Zur Beförderung der reisenden Beamten bestand eine Art Vorspannwesen, zu dessen Benutzung der Senat von Fall zu Fall eine besondere Ermächtigung erteilte. So bediente sich Cäsar, wenn er sich zum Heere begab, stets einer Tag und Nacht fahrenden Kalesche, deren Vorspann ihm gratis geleistet wurde. An mißbräuchlicher Ausnützung dieser Einrichtung fehlte es übrigens nicht. Die Senatoren z. B. verschmähten es nicht, mit Freipässen, die mehrere Jahre gültig waren, kostenfrei zu reisen. Aus dem Bisherigen erhellt, daß schon zur Zeit der Republik über das große Römische Reich ein weitverzweigtes Netz von Kommunikationsmitteln gesponnen war. So trefflich aber auch für jene Zeit diese Einrichtungen waren, es fehlte doch noch an einer einheitlichen Organisation, an einer zusammenfassenden Leitung und Überwachung der vereinzelten Institutionen. Hierzu kam es erst unter den K a i s e r n , und e r s t v o n d a a n k a n n m a n v o n e i n e m g e g l i e d e r t e n , s t a a t l i c h g e o r d n e t e n P o s t w e s e n r e d e n .
- 76. Fig. 17. Die Staatspost unter den römischen Kaisern. (Nach dem „Poststammbuch“.) Das größte Verdienst in dieser Beziehung erwarb sich gleich der erste römische Imperator, O c t a v i a n u s A u g u s t u s , durch die Errichtung des sogen. cursus publicus. Der cursus publicus war eine S t a a t s v e r k e h r s a n s t a l t , welche die Beförderungen stationsweise, mit Wechsel der Transportmittel, zu Fuß, zu Pferd oder zu Wagen sowohl für V e r s e n d u n g e n , als auch für R e i s e n wahrzunehmen hatte. Diese Einrichtung war zunächst bestimmt für die Reisen des Kaisers und seines Hofes, dann der Militärpersonen und Staatsbeamten im Dienste, der Gesandten und der zur Benutzung des cursus publicus im einzelnen Falle besonders ermächtigten Personen[43]; ferner zur Beförderung der Depeschen, Akten, Dokumente und der
- 77. Staatsgelder, sowie zum Transport von Proviant, Armatur- und Montierungsstücken, Bau-Utensilien, Kunstwerken u. s. w. Der cursus publicus beförderte demnach nicht bloß Korrespondenzen, sondern auch G e p ä c k s t ü c k e und F r a c h t e n und vor allem P e r s o n e n . P r i v a t personen und P r i v a t angelegenheiten waren von Anfang an ausgeschlossen; für Staats- und Regierungszwecke gegründet und eingerichtet, sollte er auch ausschließlich nur s o l c h e n Zwecken dienen. Jeder Kurs war in bestimmte Stationen geteilt. Solcher Stationen gab es zweierlei: solche, bei welchen bloß der Wechsel der Gespanne stattfand, und welche mutationes (vom lat. mutāre, wechseln) genannt wurden, und solche, bei welchen auch die Wagen und Postillone gewechselt wurden, und die außerdem noch zur Beherbergung der Reisenden eingerichtet waren, daher ihr Name mansiones (von manēre, bleiben) = Rastorte. Manche dieser mansiones waren sehr reichlich und schön ausgestattet. Die mansiones waren in der Regel eine Tagreise, die mutationes 1–2 Meilen voneinander entfernt. Auf jeder Mutatio mußten in der Regel 20 Zugtiere unterhalten werden, während die Mansionen deren 40 und noch mehr hatten. Die oberste Leitung des Postwesens lag seit Augustus in der Hand des Praefectus praetorio in Rom. Dies ist das Wesentlichste über den cursus publicus der Römer. Es zeigen sich daran zugleich die durchgreifenden Unterschiede von dem spätern, zuerst in Deutschland im Zeitalter der Reformation eingeführten Postwesen. So war der cursus publicus nicht für jedermann benutzbar; Beförderung fand nur statt, wenn gerade Depeschen oder Reisende vorhanden waren. Endlich war die Benutzung des cursus publicus durch die Beteiligten ganz unentgeltlich. Die empfindlichen Lasten, welche die Unterhaltung dieser Anstalt verursachte, mußte das V o l k tragen, und dafür verblieb den Provinzialen zum Troste nichts anderes, als was die Pferde in den Ställen zurückließen. Während heute die Anlegung eines Postkurses von der Gegend, durch welche er führt, als eine
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