# Questions tagged [axiom-of-choice]

An important and fundamental axiom in set theory sometimes called Zermelo's axiom of choice. It was formulated by Zermelo in 1904 and states that, given any set of mutually disjoint nonempty sets, there exists at least one set that contains exactly one element in common with each of the nonempty sets. The axiom of choice is related to the first of Hilbert's problems.

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### Does ZF + BPI alone prove the equivalence between “Baire theorem for compact Hausdorff spaces” and “Rasiowa-Sikorski Lemma for Forcing Posets”?

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### Do saturated models require choice?

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### Can we choose a sequence of Hilbert spaces?

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### Does every countable set of Turing degrees have an upper bound, without AC?

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### Applications of Zorn’s lemma that aren’t chain-complete/directed-complete?

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### Is a function needed here?

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### Does the axiom schema of collection imply schematic dependent choice in ZFCU?

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### Can we choose an element from a class?

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### Copies of the reals in $\mathbb{C}$ without the Axiom of Choice

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### A weak form of countable choice

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### A simple form of choice

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### Global choice and skeletons of large categories

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### Cofinality without choice: can this coarse definition suffer badly?

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### Do we need full choice to “efficiently” use (sub)bases?

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### Which very large cardinals are preserved under Woodin's forcing for $\mathsf{AC}$?

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### What are the known implications of “There exists a Berkeley cardinal”?

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### Does $H\vDash AC$

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### Is the hereditary version of this weak finiteness notion nontrivial?

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### Models of $\mathsf{ZF^-_2}$ over $\mathsf{ZF}$

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### Does $\text{AC}_{\text{WO}}$ prove $\Theta \neq \aleph_{\omega+1}?$

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### Is this notion of finiteness closed under unions?

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### What is first-order logic with Dedekind-finite sets of variables?

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### Do vector spaces without choice satisfy Cantor-Schroeder-Bernstein?

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### Automorphism groups of the complex numbers, and other fields

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### Wiki for consequences of axiom of choice?

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### Is the theory Flow actually consistent?

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### BCT equivalent to DC

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### Does Hahn-Banach for $\ell^\infty$ imply the existence of a non-measurable set?

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### Axiom of Countable Choice and meager sets

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### The strength of “There are no $\Pi^1_1$-pseudofinite sets”

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### Maximality principle in symmetric extensions

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### Čech functions and the axiom of choice

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### Implications of the existence of a pair of surjective functions, without Axiom of Choice

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### For which classes of metric spaces can we prove that quasi-isometry is an equivalence relation in ZF?

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### Is Axiom of Choice equivalent to its version for families of sets, indexed by ordinals? [duplicate]

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### Does the “three-set-lemma” imply the Axiom of Choice?

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### Non-uniqueness of (Galerkin) approximations and convergent subsequences without the axiom of choice?

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### A $\mathsf{ZF}$ example of a nonreflexive group which is isomorphic to its double dual?

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### Distributive lattices and axiom of choice

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### Undetermined Banach-Mazur games: beyond DC

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### Can small class choice be weaker than global choice and stronger than set choice + collection?

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### General theory of the reals in Solovay-like models

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### Is Proper Class Choice equivalent to Global Choice?

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### Models of ZF intermediate between a model of ZFC and a generic extension

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### A $\mathsf{ZF}$ example of two Baire spaces whose product is not Baire?

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### Can the axiom of choice or its weaker versions be (dis)proved using reflection principles?

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### Aronszajn Trees when AC fails

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### Very large axiom of choice

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### Does choice always hold in a model of ZF with point-wise parameter-free definable sets?

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