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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1answer
113 views

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

Rasiowa-Sikorski Lemma (for forcing posets)is the statement: For any p.o. $\mathbb{P}$ (i.e. $\mathbb{P}$ is a reflexive transitive relation) and for any countable family of dense subsets of $\mathbb{...
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Do saturated models require choice?

Let $T$ be a first-order theory, and suppose we want to build a saturated model $\mathbb U$ of $T$. That is, we want a model $\mathbb U$ of cardinality bigger than $|T|$, saturated in its own ...
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Can we choose a sequence of Hilbert spaces?

Let $n$ be a fixed natural number. Let $H$ be a complex Hilbert space and $H_1,\dotsc,H_n$ be closed subspaces of $H$. Set $H_0\mathrel{:=}H_1\cap H_2\cap\dotsb\cap H_n$ and let $P_i$ be the ...
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Does every countable set of Turing degrees have an upper bound, without AC?

It is easy to see that every countable collection of sets $A_n\subseteq\mathbb{N}$ has an upper bound in the Turing degrees, since we can just take a copy of their disjoint sum $\oplus_n A_n=\{\langle ...
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Applications of Zorn’s lemma that aren’t chain-complete/directed-complete?

Zorn’s Lemma applies to posets in which every chain has an upper bound. However, in all applications I know, the poset is also evidently chain-complete — chains have least upper bounds. A few ...
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Is a function needed here?

This question is related to my question Can we choose an element from a class?. However, I decided to create a separate question. Let $H$ be a complex Hilbert space and $H_1,\dotsc,H_n$ be closed ...
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Does the axiom schema of collection imply schematic dependent choice in ZFCU?

This question was asked on math.stackexchange and didn't receive an answer. But I think it's interesting, and I at least would love to know the answer. Let ${\sf ZFCU}$ be the axioms of ${\sf ZFC}$ ...
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Can we choose an element from a class?

Let $H$ be a complex Hilbert space and $H_1,...,H_n$ be closed subspaces of $H$. Set $H_0:=H_1\cap H_2\cap...\cap H_n$ and let $P_i$ be the orthogonal projection onto $H_i$, $i=0,1,2,...,n$. I study ...
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Copies of the reals in $\mathbb{C}$ without the Axiom of Choice

Suppose we work in a model in which the Axiom of Choice does not hold, and in which $\mathbb{C}$ only has one nontrivial automorphism (such models exist). Question: "how many" subfields of $\...
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A weak form of countable choice

Let $\Omega$ be the set/type of truth values. We're using constructive logic. Define $AC_{0, 0} = \forall P : \mathbb{N}^2 \to \Omega, (\forall n \in \mathbb{N}, \exists m \in \mathbb{N}, P(n, m)) \to ...
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A simple form of choice

While reviewing some categorical versions of the axiom of choice, it occurred to me that none of the formulations I'm aware of actually reflect how I use choice in practice: pronounce that we 'choose ...
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Global choice and skeletons of large categories

It is stated on the nlab that the axiom of choice is equivalent to the statement that all small categories have a weak skeleton, meaning a skeletal category which is equivalent to them. Is the axiom ...
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Cofinality without choice: can this coarse definition suffer badly?

This is a rephrased version of a question previously asked at MSE without success. Working in $\mathsf{ZF}$, it is no longer possible in general to give every linear order an ordinal cofinality. For ...
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Do we need full choice to “efficiently” use (sub)bases?

This question was previously asked and bountied at MSE without success. Suppose $(X,\tau)$ is a topological space, $B$ is a base for $\tau$, and $U\in \tau$ is an open set. Consider the following two ...
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Which very large cardinals are preserved under Woodin's forcing for $\mathsf{AC}$?

Woodin showed that we can force $\mathsf{AC}$ if there is a proper class of supercompact cardinals while preserving supercompacts, by forcing Easton-support iteration of $\operatorname{Col}(\kappa,<...
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What are the known implications of “There exists a Berkeley cardinal”?

Inspired by this question: What are the known implications of "There exists a Reinhardt cardinal" in the theory "ZF + j"? Definitions: $\delta$ is Berkeley iff for every $\alpha\...
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Does $H\vDash AC$

The set $H_\kappa$ of sets hereditarily of cardinality less than $\kappa$ is defined as $H_\kappa=\{x||tc(x)|\lt\kappa\}$. What if we define the set $H=H_{Ord}$ of sets hereditarily of cardinality ...
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Is the hereditary version of this weak finiteness notion nontrivial?

Say that a set $X$ is $\Pi^1_1$-pseudofinite if every first-order sentence $\varphi$ with a model with underlying set $X$ has a finite model. The existence of infinite $\Pi^1_1$-pseudofinite sets is ...
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Models of $\mathsf{ZF^-_2}$ over $\mathsf{ZF}$

Let $\mathsf{ZF^-_2}$ be a second-order ZF without Powerset, and the second-order Collection in place of Replacement. It is easy to see that every transitive model of $\mathsf{ZF^-_2}$ is closed under ...
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Does $\text{AC}_{\text{WO}}$ prove $\Theta \neq \aleph_{\omega+1}?$

The choice principle $\text{AC}_{\text{WO}}$ proves a large amount of cardinal arithmetic. It's well-known to imply DC, that successor cardinals are regular, and that for all $X$, there is $\lambda$ ...
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Is this notion of finiteness closed under unions?

This was asked and bountied at MSE without success. Throughout, we work in $\mathsf{ZF}$. Say that a set $X$ is $\Pi^1_1$-pseudofinite if for every first-order sentence $\varphi$, if $\varphi$ has a ...
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What is first-order logic with Dedekind-finite sets of variables?

The usual set up of first-order logic is with an infinite reservoir of variables which we can use in formulas. This is one of the annoying reasons why we need to put $\aleph_0$ into the cardinal ...
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Do vector spaces without choice satisfy Cantor-Schroeder-Bernstein?

If $V \hookrightarrow W$ and $W \hookrightarrow V$ are injective linear maps, then is there an isomorphism $V \cong W$? If we assume the axiom of choice, the answer is yes: use the fact that every ...
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Automorphism groups of the complex numbers, and other fields

If one accepts the Axiom of Choice (AC), then the automorphism group of $\mathbb{C}$ is a huge and wild group, very poorly understood. But apparently if one does not accept the Axiom of Choice, then ...
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Wiki for consequences of axiom of choice?

I raised the following question as part of another MO question, but I am following the suggestion of Nate Eldredge to make it a question in its own right. For many years, there has a been a valuable ...
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Is the theory Flow actually consistent?

Recently the paper Adonai S. Sant'Anna, Otavio Bueno, Marcio P. P. de França, Renato Brodzinski, Flow: the Axiom of Choice is independent from the Partition Principle, arXiv:2010.03664 appeared on ...
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BCT equivalent to DC

Do you know where I can find proof of equivalence Baire Category Theorem and DC (Axiom of Dependent Choice)? It is well known fact but I can't find appropriate literature with the proof.
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Does Hahn-Banach for $\ell^\infty$ imply the existence of a non-measurable set?

Working over ZF but without the Axiom of Choice (AC), assume that the Hahn–Banach Theorem holds for $\ell^\infty$. Does it follow that there exists a set of real numbers that is not Lebesgue ...
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Axiom of Countable Choice and meager sets

Let us recall that the Axiom of Countable Choice (denoted by ACC) says that the countable product $\prod_{n\in\omega}X_n$ of nonempty sets $X_n$ is nonempty. It is easy to see that ACC implies that ...
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The strength of “There are no $\Pi^1_1$-pseudofinite sets”

For $\Gamma$ a set of second-order sentences in the empty language, say that a set $X$ is $\Gamma$-pseudofinite if $X$ is infinite but for every sentence $\varphi\in\Gamma$ which is satisfied in every ...
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Maximality principle in symmetric extensions

Let $M$ be a ctm and $P\in M$ a forcing order. In regular forcing extensions, we have the following well-known Principle: $$p\Vdash_{M,P}\exists x\phi[x]\;\Longrightarrow\;\exists\sigma\in M^P\;p\...
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Čech functions and the axiom of choice

A Čech closure function on $\omega$ is a function $\varphi:\mathcal P(\omega)\to\mathcal P(\omega)$ such that (i) $X\subseteq\varphi(X)$ for all $X\subseteq\omega$, (ii) $\varphi(\emptyset)=\emptyset$,...
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Implications of the existence of a pair of surjective functions, without Axiom of Choice

The classical Cantor-Schroder-Bernstein Theorem says that there exists a bijective function $X\leftrightarrow Y$ if and only if there exist injective functions $X\hookrightarrow Y$ and $Y\...
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For which classes of metric spaces can we prove that quasi-isometry is an equivalence relation in ZF?

Given two metric spaces $(M_1, d_1)$ and $(M_2, d_2)$, a map $\phi \colon (M_1, d_1) \to (M_2, d_2)$ is a large-scale Lipschitz essentially surjective map if there exist constants $A \geq 1, B \geq 0$,...
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Is Axiom of Choice equivalent to its version for families of sets, indexed by ordinals? [duplicate]

Is Axiom of Choice equivalent to the following statement? Axiom of Ordinal Choice: For any ordinal $\lambda$ and any indexed family of sets $(X_\alpha)_{\alpha\in\lambda}$ there exists a function $...
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Does the “three-set-lemma” imply the Axiom of Choice?

Consider the following curious statement: $(S)$ $\;$ Let $X$ be a non-empty set and let $f:X \to X$ be fixpoint-free (that is $f(x) \neq x$ for all $x\in X$). Then there are subsets $X_1, X_2, X_3 \...
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Non-uniqueness of (Galerkin) approximations and convergent subsequences without the axiom of choice?

Suppose I have an equation in some reflexive separable Banach space $X$: $$Au=f$$ for given data $f \in X^*$ and $A\colon X \to X^*$ a pseudo-monotone operator. Existence can be proved via Galerkin ...
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A $\mathsf{ZF}$ example of a nonreflexive group which is isomorphic to its double dual?

Given a group $G$ denote by $G^\ast=\mathrm{Hom}(G,\Bbb Z)$ its dual and by $j\colon G\to G^{\ast\ast}$ the canonical homomorphism $g\mapsto (f\mapsto f(g))$. A group is reflexive iff $j$ is an ...
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Distributive lattices and axiom of choice

What form of the axiom of choice is equivalent (in ZF) to the statement that every distributive lattice is isomorphic to a lattice of sets?
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Undetermined Banach-Mazur games: beyond DC

This question is a follow-up to this one; see that question for the definition of Banach-Mazur games. There James Hanson showed that ZF+DC proves that there is an undetermined Banach-Mazur game; ...
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Can small class choice be weaker than global choice and stronger than set choice + collection?

In this posting what was termed as "Proper Class Choice" principle turned to be equivalent to Global Choice over the base theory of "MK-Foundation -Limitation of size + Set Replacement*&...
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General theory of the reals in Solovay-like models

Solovay's model is a famous model of $\sf ZF$ where we start in $L$ with $\kappa$ inaccessible, and we collapse all the ordinals below $\kappa$ to be countable, without collapsing $\kappa$ itself, and ...
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Is Proper Class Choice equivalent to Global Choice?

Working in "MK-Regularity-Limitation of Size + Replacement for sets", call it the Base theory, let's coin the following axiom: Axiom of Super-Choice:$$\forall \ relation \ R \ \exists F \...
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Models of ZF intermediate between a model of ZFC and a generic extension

Let $M$ be a countable model of $ZFC$ and $M[G]$ be a (set) generic extension of $M$. Suppose $N$ is a countable model of $ZF$ with $$M\subseteq N \subseteq M[G]$$ and that $N=M(x)$ for some $x\in ...
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A $\mathsf{ZF}$ example of two Baire spaces whose product is not Baire?

Motivated by this question I'm looking for a pair of Baire topological spaces whose product is not Baire and whose construction does not need the axiom of choice. The example of such spaces I'm ...
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1answer
206 views

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

In On the Question of Absolute Undecidability, Peter Koellner investigates whether it is possible to prove or disprove $V = L$ using (EDIT: both first and second-order) reflection principles, ie. ...
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Aronszajn Trees when AC fails

This question may be easy and indicative of my ignorance about the failure of the axiom of choice. If so, I apologize. Below assume $\mathsf{DC}$ but not $\mathsf{AC}$. Suppose we have a partial order ...
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1answer
591 views

Very large axiom of choice

let me say that I am not a set theorist, but I have to settle up some things in category theory and I need your help. What I'd like to do is, in some way, use axiom of choice for proper classes. I ...
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1answer
236 views

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

If one add to ZF the rule that all sets are parameter free definable. Would that prove the axiom of choice? More specifically. IF we add the following omega rule to inference rules of the language of ...
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1answer
338 views

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

It is a well-known fact that closed convex sets in Banach spaces are weakly closed. The common proof is based on the Hahn-Banach theorem that uses the axiom of choice. Is there any proof of this fact ...

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