Enhanced Set Theory

Enhanced Set Theory
the deﬁnition and its use

Note for the Novice
Believe nothing, no matter where you read it, or who said it, no
matter if I have said it, unless it agrees with your own reason and
your own common sense.
— Gautama Buddha

Introduction
Here we introduce an Enhanced Set Theory, by adding to ZFC
theory a Universal Number System and an associated Axiom of
Inﬁnitesimals, the motivation being that we can have a simple
visualization of the real line without going beyond the cardinal ℵ0.

Universal Number System
A binary number is usually defined as a two way infinite binary
sequence around the binary point,
. . . 000xx . . . xxx•xxxxx . . .
in which the xs represent either a 0 or 1, and the infinite sequence
on the left eventually ends up in 0s. The two’s complement number
system represents a negative number by a two way infinite sequence,
. . . 111xx . . . xxx•xxxxx . . .
in which the infinite sequence on the left eventually ends up in 1s.
In this representation, the right sequence is always nonterminating.

For example, the sequence,
. . . 000010011•10101010 . . .
represents the number 19.6666 · · · . Similarly, the complement of
this sequence,
. . . 111101100•01010101 . . .
represents the negative number −19.6666 · · · . Note the restriction
in our deﬁnition of a real number: the left sequence must
eventually end up in either all 1s or all 0s.

We define the Universal Number System as the number system
in which there are no restrictions on the infinite sequence on the
left side. It is easy to recognize that, the sequence
. . . 00000•xxxxx . . .
with a nonterminating binary sequence on the right side represents
a number in a unit interval. The two way infinite sequence we get
when we flip the sequence around the binary point,
. . . xxxxx•00000 . . .
we call a supernatural number.

Extending the idea a little further we call the number
. . . xxxxx•xxxxx . . .
with infinite sequences on both sides, a super number or a black
stretch. The infinite set of black stretches, we define as the
blackwhole. Without getting into details, we may consider the
blackhole as a dual of the finite part of the real line, for which
reason, we may call the finite part of the real line the whitewhole.
The name black stretch is supposed to suggest that it is a set of
points distributed over an infinite line. Our description of the
blackwhole clearly indicates that it can be used to visualize what is
beyond the finite physical space around us.

Axiom of Inﬁnitesimals
If k is a cardinal, we write
ℵ0
k
for the cardinality of the set of all subsets of ℵ0 with the same
cardinality as k.

It is easy to see that an inﬁnite subset of positive integers can be
used to represent a point in the interval (0,1]. For example, the
inﬁnite binary sequence
•10011111 . . .
has the representation {1, 4, 5, 6, . . .}, where we have picked those
positions where the 1s occur. From this it should be clear that the
cardinality of the set of points within a unit interval is
ℵ0
ℵ0
= 2ℵ0
,
a known result.

We have seen that if we consider all conceivable subsets of positive
integers, we get the cardinality of set of points inside the unit
interval as 2ℵ0
. A simplified set theory results, if we choose to
consider only infinite recursive subsets of positive integers and call
the corresponding elements in the unit interval, infinitesimals. We
use the symbol r for a recursive subset and R for the entire set of
recursive subsets. It is well-known in recursive function theory that
the cardinality of R is ℵ0. Incidentally, recursive sets are those sets
for which clear algorithms exist. Since algorithms can be
enumerated, it follows that the cardinality of R is ℵ0.

First of all, let us note that corresponding to every real recursive
number it is possible to visualize an infinitesimal attached to it. We
will illustrate this with an example. Consider the number 2/3
written as an infinite sequence 0.101010... and its finite
terminations 0.1, 0.101, 0.10101, ... which can be used to represent
the intervals (1/2, 2/3), (5/8, 2/3), (21/32, 2/3), ... respectively.
Note that the length of the interval decreases monotonically when
the length of the termination increases and the cardinality of the
set of points inside these intervals remain constant at 2ℵ0
.

Further we notice that the intervals
(1/2, 2/3),
(5/8, 2/3),
(21/32, 2/3),
...
can also be represented explicitly by the sequences
0.1xxx...,
0.101xxx...,
0.10101xxx...,
...
where the x’s represent either a 0 or 1.

From this, we can say that an infinitesimal is what we get when we
visualize the interval corresponding to the entire nonterminating
sequence, and this infinitely small interval contains 2ℵ0
points in it.
The infinitely small interval derived from the recursive number r
containing the 2ℵ0
points, we represent by the notation
ℵ0
ℵ0 r
.
Also, we write
ℵ0
ℵ0 r
r ∈ R =
ℵ0
ℵ0 R
.

Axiom of Infinitesimals:
(0, 1] =
ℵ0
ℵ0 R
where
ℵ0
ℵ0 r
is an infinitesimal representing r. The axiom of infinitesimals makes
it easy to visualize the unit interval (0, 1]. The axiom splits the unit
interval into a set of infinitesimals with cardinality ℵ0.

Infinitesimal Graph
Since every infinitesimal separates the recursive numbers (the
numbers corresponding to recursive sets) within the unit interval
into two mutually exclusive sets, it may not be unreasonable to call
each infinitesimal a dedekind edge. With this terminology, we can
visualize the unit interval as an infinitesimal graph with dedekind
edges in it. From what we have said before, it is clear that the
cardinality of the set of edges in a unit interval is ℵ0.

Conclusion
In enhanced set theory (EST) we have a provision to restrict
ourselves to two kinds of numbers. The ﬁrst one is the natural
numbers which provide us the unit intervals. The set of natural
numbers has the cardinality ℵ0. The second one is the recursive
numbers which provide us the inﬁnitesimals on the real line. The
set of recursive numbers also has the cardinality ℵ0.

EST, while accepting the reals within an infinitesimal, allows us to
ignore them and deal only with the infinitesimals and the cardinal
ℵ0, when necessary. The beneficiaries here are the scientists who
may not be particularly interested in Lebesgue measure, inaccessible
cardinals, Skolem paradox, and such other esoteric issues. It is easy
to see that scientists will still have the Riemann integral, be able to
send a man to the moon and carryout their nuclear reactor design.

Enhanced Set Theory

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