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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.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The Axiom of Choice and Zorn's lemma are equivalent
Statement
Over ZF, the Axiom of Choice (The Axiom of Choice) and Zorn's lemma (Zorn's lemma) are equivalent: each implies the other.
Facts & Assumptions
Given: The axioms of ZF.
The Axiom of Choice implies Zorn's lemma (Zorn's lemma).
Zorn's lemma implies the Axiom of Choice (Zorn's lemma implies the Axiom of Choice).
Proof
Assuming the Axiom of Choice, every nonempty poset in which each chain has an upper bound has a maximal element, which is Zorn's lemma.
Assuming Zorn's lemma, every family of nonempty sets has a choice function, which is the Axiom of Choice.
Each statement implies the other over ZF, so they are equivalent.
Remarks
- This is the item later pages cite when they want to use either form without re-arguing the passage between them. The ultrafilter lemma (The ultrafilter lemma, from the Axiom of Choice: every filter extends to an ultrafilter ↗) uses the Zorn form; results about products of nonempty sets use the choice-function form.
- Equivalence is over ZF, and it is a genuine two-way implication proved here, not an appeal to authority. What is not proved here, and cannot be until forcing is available, is that either statement is independent of ZF. That rests on two external results this library records but does not prove, Gödel 1938: ZF does not refute the Axiom of Choice ‡ and Cohen 1963: ZF does not prove the Axiom of Choice ‡; where the weaker choice principles sit is What the ultrafilter lemma costs: a choice principle strictly weaker than AC ↗, and the corresponding trap is FALSE: Zorn's lemma is a theorem of ZF.
- Because the two are equivalent, a theorem proved with Zorn's lemma costs at most the Axiom of Choice, and the equivalence says nothing beyond that. It does not say the theorem needs the Axiom of Choice: a proof through Zorn is an upper bound on the price, never a lower one. The ultrafilter lemma (The ultrafilter lemma, from the Axiom of Choice: every filter extends to an ultrafilter ↗) is the standing example, proved here with Zorn and yet, if ZF is consistent, on the external results recorded in What the ultrafilter lemma costs: a choice principle strictly weaker than AC ↗, strictly weaker than the Axiom of Choice, so that proof overpays. The consistency hypothesis is not decoration and cannot be dropped: "strictly weaker" is a relative-consistency claim, resting on models of ZF in which the ultrafilter lemma holds and the Axiom of Choice fails, and an inconsistent ZF would prove everything, collapsing the separation. Only a statement that is itself equivalent to the Axiom of Choice, as Zorn's lemma is by this corollary, costs exactly the Axiom of Choice, no more and no less. Where the weaker principles sit is What the ultrafilter lemma costs: a choice principle strictly weaker than AC ↗.
Depends on
Used by
- Choice, Zorn and well-ordering are equivalent Corollary
- FALSE: Zorn's lemma is a theorem of ZF False statement
- The choice ledger: what costs the Axiom of Choice and what does not Remark
- What the ultrafilter lemma costs: a choice principle strictly weaker than AC Remark
- Alexander's subbase lemma: if every cover by members of a fixed subbasis has a finite subcover then the space is compact; the proof is an application of Zorn's lemma Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 16 results over 6 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
- Encyclopedia of Mathematics, Zorn lemma (standard reference, not scraped)
- I. Khatchatourian, The Axiom of Choice (University of Toronto MAT327 notes) (standard reference, not scraped)
- Zorn's lemma (Wikipedia) (standard reference, not scraped)
- Axiom of choice (Wikipedia) (standard reference, not scraped)