The Relative Consistency of the Axiom of Choice — Mechanized Using Isabelle/ZF
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This document discusses Lawrence C. Paulson's work mechanizing the proof of the relative consistency of the axiom of choice using the Isabelle/ZF theorem prover. It summarizes key aspects of Gödel's original proof, outlines how concepts like the constructible universe and satisfaction are defined in Isabelle/ZF, and describes proving that L satisfies the ZF axioms and that V=L holds in L. The work resulted in a fully mechanized proof of consistency for the axiom of choice without relying on formalized metamathematics.
The Relative Consistency of the Axiom of Choice — Mechanized Using Isabelle/ZF
- 1. The Relative Consistency of the Axiom of Choice Mechanized Using Isabelle/ZF Lawrence C. Paulson Computer Laboratory
- 2. Why Do Proofs By Machine? • Too many been done already! – Gödel’s incompleteness theorem (Shankar) – thousands of Mizar proofs • But many types of reasoning are hard to formalize. – Algebraic structures (e.g. group theory) – Proofs involving metamathematics • And this one concerns Hilbert’s First Problem! 2
- 3. Outline of Gödel’s Proof • • • • Define the constructible universe, L Show that L satisfies the ZF axioms Show that L satisfies the axiom V=L Show that V=L implies AC and GCH A contradiction from ZF and V=L can be translated into one from ZF alone. 3
- 4. The Sets That Must Exist 4
- 5. L satisfies the ZF axioms • Union, pairing – Unions and pairs are definable by formulae • Powerset, replacement scheme – Using a rank function for L • Comprehension scheme (separation) – By the Reflection Theorem – Scheme can be proved only in the metatheory 5
- 6. Show that L satisfies V=L • V=L means “all sets are constructible” • The concept of “constructible” is absolute • Absolute means same in all models – Most concepts are absolute: unions, ordinals, functions, bijections, etc. – Not absolute: powersets, function spaces, cardinals 6
- 7. Show that V=L implies AC (or rather, the well-ordering theorem) • The set of formulae is countable • Parameter lists for formulae can be wellordered lexicographically • So, if X is well-ordered then so is D(X) • Inductively construct a well-ordering on L 7
- 8. Satisfaction for Class Models? For M a set, can define satisfaction recursively: For M a class, satisfaction cannot be defined! The nondefinability of truth (Tarski) 8
- 9. Satisfaction Defined Syntactically The relativization of f to M 9
- 10. A contradiction using V=L? • Can prove that (V=L)L is a ZF theorem • … as is f L provided f is a ZF axiom • Thus, a contradiction from ZF + (V=L) amounts to a contradiction in ZF alone • Developing the argument (Gödel never did) requires proof theory 10
- 11. Isabelle/ZF • Same code base as Isabelle/HOL • Higher-order metalogic, ideal for – Theorem schemes – Classes – Class functions • Develops set theory from the ZermeloFraenkel axioms to transfinite cardinals 11
- 12. Defining the Class L in Isabelle • Datatype declaration of the set formula • Primitive recursive functions: – Satisfaction relation – Arity of a formula – De Bruijn renaming • Definable powersets: Dpow(X) • Constructible hierarchy: Lset(i) • The predicate L 12
- 13. Relativization in Isabelle • Define a separate predicate for each concept: 0, », «, function, limit ordinal, … • Make each predicate relative to a class M • Absoluteness: prove that the predicate agrees with the native concept Outcome: a relational language of sets 13
- 14. Examples: Pairs and Domains 14
- 15. Proving that L is a Model of ZF • Express ZF axioms using the predicates • Mechanize proofs from Kunen (1980) • Separation axiom (comprehension): – By previous proof of Reflection Theorem – Meta-$ quantifier to hide giant classes – Automatic translation from real formulae to elements of the set formula – 40 separate instances proved 15
- 16. Proving that L is a Model of V=L • Absoluteness of well-founded recursion • Absoluteness and relativization for … – Recursive datatypes – About 100 primitive concepts – The satisfaction function (detailed breakdown needed) • The concepts Dpow(X) and Lset(i) • Define Constructible(M,x) • Finally prove L(x) fi Constructible(L,x) 16
- 17. Comparative Sizes of Theories (in Tokens) Reflection theorem 3400 Definition of L 4140 ZF holds in L (excluding separation) 5100 V=L holds in L 29700 V=L implies AC 1769 17
- 18. Doing without Metamathematics • Can’t reason on the structure of formulae • Can’t prove separation schematically • Can’t formalize how a contradiction from V=L leads to a contradiction in ZF • But: can use native set theory – Isabelle/ZF’s built-in set theory libraries – benefits of a shallow embedding 18
- 19. Conclusions • • • • A mechanized proof of consistency for AC Big:12000 lines or 49000 tokens Just escape having to formalize metatheory Future challenges: – Repeat, with a formalized metatheory – Prove generalized continuum hypothesis – Formalize forcing proofs: independence of AC 19