# 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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### 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?

**7**

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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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### Are closed convex subsets of a Banach space weakly closed without the axiom of choice?

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### How much choice is required for a countably-infinite index subgroup of the real additive group?

**14**

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### Complement-like operator and the axiom of choice

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### Uncountable chain of nested sets without choice

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### Can the set of endomorphisms of $(\mathbb R,+)$ have cardinality strictly between $\frak c$ and $\frak{c^c}$?

**5**

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### Failure of SVC in Grothendieck toposes

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### Existence of a certain set of 0/1-sequences without the Axiom of Choice

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### Polish transversals

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### Exterior powers and choice

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### Relationship between AC, WO, and Zorn's lemma in ZF-Powerset

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### Injection into a proper class and choice without regularity

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### Measure of rational hyperplanes of $\mathbb{R}$

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### Cardinal characteristics of amorphous sets

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### Amorphous proper classes in MK

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### How many algebraic closures can a field have?

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### Is Global Choice conservative over Zermelo with Choice?

**10**

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### Axiom of choice and algebraic tensor product

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### Is this lemma equivalent to the axiom of choice?

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### Maximizing “happy” vertices in splitting an infinite graph

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### Surreal numbers and the Axiom of Choice

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### Stronger negation of AC given by rejecting “infinite hat” puzzles

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### Axiom of Choice versus V=L in opposition to large cardinals

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### Compactification of Tychonoff spaces without full axiom of choice

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### Relation between the Axiom of Choice and a the existence of a hyperplane not containing a vector

**14**

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### Cardinality vs. isomorphism type of vector spaces without choice

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### Can we have an infinite sequence of decreasing cardinality all terms of which have equal sized power sets?

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### Are there known examples of sets whose power set is equal in size to power set of larger sets only in absence of choice?

**14**

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### Do choice principles in all generic extensions imply AC in $V$?

**10**

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### The Tall Tale of Terminating Transfinite Towers

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### The partial preorder on $\mathbb N$ generated by the finite axioms of choice

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### The Rise and Fall of Dictators & How it Depends on Our Choice

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