Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (openai/gpt-5.4)verified 2026-07-26 (claude-opus-5)‡ rests on unproved material
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced — the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted — a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated — a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 1 statement not proved in this library. Every dependency marked ‡ below is recorded with a citation but is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

FALSE: Zorn's lemma is a theorem of ZF

Statement

FALSE. Zorn's lemma is a theorem of ZF: it can be proved from the Zermelo–Fraenkel axioms without assuming the Axiom of Choice.

The statement is plausible because Zorn's lemma reads like a structural fact about ordered sets rather than a selection principle, and because the proof given in Zorn's lemma runs through Bourbaki–Witt fixed point theorem, which genuinely is choice-free. The Axiom of Choice enters that proof at a single step, and it cannot be removed.

Facts & Assumptions

Given: The axioms of ZF, assumed to be consistent, together with the external metamathematical result cited below. Every conclusion here is relative to that consistency assumption, which cannot be dropped and cannot be proved inside ZF.

‡ [A1]

If ZF is consistent, then ZF does not prove the Axiom of Choice (Cohen 1963, Cohen 1963: ZF does not prove the Axiom of Choice ‡). This is an external result, established by forcing and quoted rather than proved here; it presupposes the consistency of ZF assumed in the Given. See the remark below.

[L1]

Zorn's lemma implies the Axiom of Choice, and this implication is itself proved in ZF (Zorn's lemma implies the Axiom of Choice).

[L2]

The two statements are equivalent over ZF (The Axiom of Choice and Zorn's lemma are equivalent).

Refutation

technique · contradiction
1.1

Suppose Zorn's lemma were a theorem of ZF.

assume-contra
1.2

The implication from Zorn's lemma to the Axiom of Choice is proved in ZF, using no choice principle.

L1L2
2.1

Chaining a ZF theorem with a ZF-provable implication yields a ZF theorem, so the Axiom of Choice would be a theorem of ZF.

step 1.1step 1.2
3.1

This contradicts [A1], which holds under the consistency of ZF assumed in the Given; so, under that assumption, Zorn's lemma is not a theorem of ZF. Equivalently and without any assumption: if ZF proves Zorn's lemma, then ZF proves the Axiom of Choice, and ZF is therefore inconsistent.

step 2.1‡ A1discharge-contradiction∎

Remarks

  • What is and is not proved here. The refutation is a genuine ZF argument given the cited independence result, but that result is quoted rather than proved here: Cohen's theorem requires forcing. The honest reading is therefore conditional, namely that Zorn's lemma is a theorem of ZF only if ZF is inconsistent. It is recorded this way deliberately rather than presented as fully derived.
  • The companion half of the independence, that ZF cannot refute the Axiom of Choice, is Gödel's 1938 constructible universe result. Together they say the Axiom of Choice, and hence Zorn's lemma, is genuinely independent.
  • Where this sits among the choice principles, and which weaker ones are still unprovable in ZF, is taken up later in What the ultrafilter lemma costs: a choice principle strictly weaker than AC ↗.
  • The trap this item exists to close: Bourbaki–Witt fixed point theorem really is choice-free and does most of the work of Zorn's lemma, which invites the conclusion that the whole proof is choice-free. Step 4.1 of Zorn's lemma, where a strict upper bound is selected for every chain at once, is the irreducible use.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 22 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources