The document discusses the mechanization of set theory using the Isabelle proof assistant, focusing on cardinal arithmetic and the axiom of choice. It presents proofs related to well-ordering theorems and the cardinality of sets, alongside challenges encountered in the mechanization process. It highlights the aim to justify recursive definitions and outlines obstacles such as uneven gaps in human proofs and differing definitions of concepts.

1. Gra¸bczewski & Paulson Mechanizing Set Theory 1
Mechanizing Set Theory:
Cardinal Arithmetic and the Axiom of Choice
Krzysztof Gra¸bczewski, Copernicus University, Torun, Poland
Lawrence C Paulson, Computer Laboratory, Cambridge University, UK
Funding: EPSRC grant GR/H40570; TEMPUS Project JEP 3340; ESPRIT Project 6453

2. Gra¸bczewski & Paulson Mechanizing Set Theory 2
The Generic Proof Assistant Isabelle
many logics higher-order syntax unification
• Expressions are typed λ-terms
• Schematic rules are generalized Horn clauses (like λProlog’s)
• Resolution applies rules for proof checking
• Tactic language allows user-defined automation
• Generic packages include simplifier, tableau prover, ...

3. Gra¸bczewski & Paulson Mechanizing Set Theory 3
Some Isabelle Logics
• FOL, Constructive Type Theory, modal logics, linear logic, ...
• ZF set theory
– Built upon FOL
– Lamport’s Temporal Logic of Actions (Sara Kalvala)
– Milner & Tofte’s co-induction example (Jacob Frost)
• HOL
– I/O Automata (Nipkow & Slind)
– hardware examples (Sara Kalvala)
– semantic equivalence (L¨otzbeyer & Sandner)

4. Gra¸bczewski & Paulson Mechanizing Set Theory 4
The Cardinal Proofs
• Aim: justify recursive definitions like D = 1 + D + (ω → D)
• Basis: theories of relations, functions, recursion, ordinals, ...
• Method: mechanize most of Kunen, Set Theory, Chapter I.
– orders
– order-isomorphisms
– order types
– ordinal arithmetic
– cardinality
– infinite cardinals
– AC

5. Gra¸bczewski & Paulson Mechanizing Set Theory 5
Kunen’s Proof of κ ⊗ κ = κ
“By transfinite induction on κ. Then for α < κ, |α × α| = |α| ⊗ |α| < κ.
Define a wellordering on κ × κ by α, β γ, δ iff
max(α, β) < max(γ, δ) ∨ [max(α, β) = max(γ, δ) ∧
α, β precedes γ, δ lexicographically].
Each α, β ∈ κ × κ has no more than
|(max(α, β)) + 1 × (max(α, β)) + 1| < κ
predecessors in , so type(κ × κ, ) ≤ κ, whence |κ × κ| ≤ κ. Since clearly
|κ × κ| ≥ κ, |κ × κ| = κ.”

6. Gra¸bczewski & Paulson Mechanizing Set Theory 6
Formulations of the Well-Ordering Theorem
W O1: Every set can be well-ordered.
W O2: Every set is equipollent to an ordinal number.
...
W O6: For every set x, there exists m ≥ 1, an ordinal α, and a function f
defined on α such that f (β) m for every β < α and β<α f (β) = x.
W O7: For every set A, A is finite ⇐⇒ for each well-ordering R of A,
also R−1 well-orders A.
From Rubin & Rubin, Equivalents of the Axiom of Choice, Chapter 1

7. Gra¸bczewski & Paulson Mechanizing Set Theory 7
Formulations of the Axiom of Choice
AC1: If A is a set of non-empty sets then there exists f such that
f (B) ∈ B for all B ∈ A.
...
AC6: The product of a set of non-empty sets is non-empty.
...
AC16(n, k): If A is an infinite set then there is a set tn of n-element subsets
of A such that each k-element subset of A is a subset of exactly one
element of tn. (1 < k < n)
From Rubin & Rubin, Equivalents of the Axiom of Choice, Chapter 2

8. Gra¸bczewski & Paulson Mechanizing Set Theory 8
Proof of W O6 ⇒ W O1
Lemma. If W O6 and y × y ⊆ y then y can be well-ordered.
Proof: by induction using Lemma (ii) below.
Theorem. If W O6 then every set x can be well-ordered.
Proof: Define y such that x ⊆ y and y × y ⊆ y.
y =
n∈ω
zn, where



z0 = x
zn+1 = zn ∪ (zn × zn)
Hence x is a subset of a well-ordered set.

9. Gra¸bczewski & Paulson Mechanizing Set Theory 9
Lemma for W O6 ⇒ W O1
Let Ny = m : ∃ f,α dom( f ) = α, β<α f (β) = y, ∀β<α f (β) m
Lemma (ii): If m ∈ Ny and m > 1 then m − 1 ∈ Ny.
Proof: Assume y × y ⊆ y and m ∈ N(y). Then f and α exist. Put
uβγ δ
def
= [ f (β) × f (γ )] ∩ f (δ) (β, γ, δ < α)
Clearly uβγ δ m, dom(uβγ δ) m, rng(uβγ δ) m.
Case 1: ∀β<α. f (β) = 0 → ∃γ,δ<α. dom(uβγ δ) = 0 ∧ dom(uβγ δ) m
Case 2: ∃β<α. f (β) = 0 ∧ ∀γ,δ<α. dom(uβγ δ) = 0 → dom(uβγ δ) ≈ m
Complex reasoning reduces m (and doubles α) in both cases.

10. Gra¸bczewski & Paulson Mechanizing Set Theory 10
Observations
• Mechanisation of parts of two advanced texts
– Kunen, Set Theory, most of Chapter I (Paulson)
– Rubin & Rubin, Equivalents of AC, Chapters 1–2 (Gra¸bczewski)
• Obstacles to faithful mechanisation
– unevenly-sized gaps in human proofs (intuitive leaps)
– different definitions of standard concepts
• Features for future systems?
– type inclusions, e.g. naturals ⊆ cardinals ⊆ ordinals ⊆ sets
– inheritance of structure (for algebra)