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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7 votes
0 answers
163 views

"Minimal-ish" Dedekind-finite cardinalities of models

Throughout, we work in $\mathsf{ZF}+$ "There is an infinite Dedekind-finite set." Say that a Dedekind-finite cardinality $\kappa$ is $\Sigma^1_1$-isolated iff there is some first-order ...
3 votes
2 answers
201 views

Posets such that the collection of principal down-sets does not have property ${\bf B}$

We say that a hypergraph $H=(V,E)$ has property ${\bf B}$ if there is $S\subseteq V$ such that for all $e\in E$ with $|e|>1$ we have $S\cap e \neq \emptyset \neq e \setminus S$. Let $(P,\leq)$ be a ...
1 vote
0 answers
722 views

Finding a unique and finite expected value for almost all measurable functions?

Let $(X,d)$ be a metric space. If set $A\subseteq X$, let $H^{\alpha}$ be the $\alpha$-dimensional Hausdorff measure on $A$, where $\alpha\in[0,+\infty)$ and $\text{dim}_{\text{H}}(A)$ is the ...
4 votes
0 answers
146 views

The monochromatic principle and the axiom of choice

For any set $A\neq\emptyset$, denote by $[A]^A$ the collection of sets $B\subseteq A$ such that there is a bijection $\varphi:B\to A$. If ${\cal S}\subseteq [A]^A$, we say that $B\in[A]^A$ is ...
5 votes
1 answer
126 views

References for the axiom of surjective comparability

The axiom $W_\kappa$, for $\kappa$ a cardinal, is the statement that for all sets $X$, either $|X|\leq\kappa$ (that is, there is an injection $X\to\kappa$) or $\kappa\leq|X|$. Is there literature on ...
3 votes
0 answers
165 views

Do the difficulties in generalising Henstock-Kurzweil still exist if every subset of $\mathbb R^n$ is Lebesgue measurable?

There are apparently some difficulties generalising the Henstock-Kurzweil integral from functions of signature $\mathbb R\to\mathbb R$ to functions of signature $\mathbb R^n \to \mathbb R$. One ...
8 votes
1 answer
238 views

Non-Ramsey functions $c:[\omega]^\omega\to\{0,1\}$ and the Axiom of Choice

Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$, and let $c:[\omega]^\omega\to\{0,1\}$ be a function. We say that $a\in [\omega]^\omega$ is monochromatic with respect to $c$...
7 votes
2 answers
593 views

Involutions in the absolute Galois group (and the Axiom of Choice)

It is known that the only elementary abelian $2$-groups (finite and nonfinite) in $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ are in fact finite and cyclic – that is to say, they are of order $2$....
10 votes
2 answers
650 views

New Foundations and weaker forms of choice

New Foundations (introduced by Quine) proves that $AC$ is false. Out of curiosity, is $NF$ consistent with countable choice or dependent choice? What's the strongest consequence of choice still ...
4 votes
0 answers
218 views

stating large cardinal axioms in ZF

Can I ask whether there is a good reference for how to state the standard large cardinal axioms in the context of $ZF$? My concern is that it seems that the usual proof that embeddings defined from ...
9 votes
1 answer
612 views

Small Violations of Choice: Can we force AC without collapsing the cardinalities of ordinals?

We say that a model $M$ of $\mathsf{ZF}$ satisfies Small Violations of Choice ($\mathsf{SVC}$) if all (any) of the following apply: There is a model $V\subseteq M$ such that $V\vDash\mathsf{ZFC}$, ...
14 votes
1 answer
1k views

Stone-Weierstrass Theorem without AC

To what extent does the usual Stone-Weierstrass Theorem depend on some form of the Axiom of Choice? There seems to be a lot of literature on constructive versions in toposes, but I have been unable ...
3 votes
0 answers
178 views

Basic cardinal arithmetic without choice

Do we know everything about addition and multiplication of cardinalities in choiceless set theory? For example, let $M$ be a model of $\textsf{ZF}+\textsf{AD}+V=L(\mathbb{R})$, consider the sets $\...
2 votes
0 answers
112 views

Elementary abelian 2-subgroups of $\mathrm{Aut}(\overline{\mathbb{Q}}/\mathbb{Q})$ (with and without choice)

Consider the absolute Galois group $G_{\mathbb{Q}} := \mathrm{Aut}(\overline{\mathbb{Q}}/\mathbb{Q})$. As I understand it, the only torsion elements have order $2$ (by Artin-Schreier), and they are ...
7 votes
0 answers
307 views

What is the strongest form of the Axiom of Choice available in $\mathsf{Z}_{2}$?

$\mathsf{Z}_{2}$ denotes second-order arithmetic. Some forms of AC are expressible in $\mathsf{Z}_{2}$; for example the $\mathsf{\Sigma}_{1}^{1}$ axiom of choice is part of the theory $\mathsf{ATR}_{0}...
15 votes
1 answer
1k views

In ZF, when is a disjoint union of metrizable spaces metrizable?

It is easy to see that the disjoint union $\bigsqcup_i X_i$ of a collection of metric spaces is metrizable, simply by rescaling or chopping off the individual metrics to have diameter at most one, and ...
5 votes
1 answer
472 views

Large cardinals in ZF + DC + AD

The Axiom of Dependent Choice (DC) is often considered to be an "intuitive and non-controversial" version of choice used in the proofs of many theorems in Analysis. Similarly, the Axiom of ...
6 votes
1 answer
477 views

Automorphisms of algebraically closed fields without the Axiom of Choice

In the paper Algebraische Konsequenzen des Determiniertheits-Axioms (U. Felgner – K. Schulz, Arch. Math. (Basel) 42 (1984), pp. 557–563), the authors show that in models of Zermelo-Fraenkel set theory ...
6 votes
0 answers
154 views

Weak trichotomy principle in the absence of choice

It is well known that the trichotomy property of cardinals ($∀κ,λ\in\operatorname{Card}(κ<λ∨κ=λ∨κ>λ)$) is equivalent to the axiom of choice. D. Feldman and M. Orhon had defined in [1] a ...
4 votes
1 answer
311 views

Automorphisms of vector spaces and the complex numbers without choice

In Zermelo-Fraenkel set theory without the Axiom of Choice (AC), it is consistent to say that there are models in which: there are vector spaces without a basis; the field of complex numbers $\mathbb{...
-2 votes
1 answer
196 views

Does cardinal definable choice imply AC?

Recall the definition of cardinal definable sets, to re-iterate: $Define: X \text { is cardinal definable} \iff \\\exists \text { cardinal } \kappa \, \exists \text { cardinals } \lambda_1,.., \...
4 votes
1 answer
196 views

Exactly how much (and how little) can partial ordered sets (classes) embed to the cardinalities

In the paper "Convex Sets of Cardinals", Truss mentioned a result of Jech: If $M$ is a countable transitive model of ZFC, and $(P,<)∈M$ is a poset, then there exists a Cohen extension of ...
4 votes
1 answer
645 views

Does Tarski's squaring theorem imply Axiom of Choice in NFU?

I'm trying to see which results from mainstream set theory (ZF) about Axiom of Choice can be proved in New Foundations with Urelements (U is added simply because ...
1 vote
2 answers
158 views

Does n-well ordered choice schema imply the axiom of choice?

Define: $\operatorname {wo}^n(x) \iff \forall y (y \in^n x \to \operatorname {wo} (y))$ Where $\operatorname {wo}(y)$ refers to $y$ being well orderable. Where $y \in^0 x \iff y=x \\ y \in^{n+1} x \...
1 vote
0 answers
76 views

Is this version of Nested Selection equivalent to AC?

This is an endeavor to salvage "Nested Selection" principle presented in a prior posting. Define $ \begin{align} Y \text { is } \Phi \text{-image of } X \iff &\forall a \in X \exists b \...
-4 votes
1 answer
263 views

Is Nested Selection equivalent to AC?

Nested Selection: For every infinite set $G$ of pairwise disjoint infinite sets such that any two distinct elements $x,y$ of $G$ either "$y$ is a set of proper supersets of elements of $x$ and ...
7 votes
0 answers
193 views

Does the Weak Power Hypothesis imply the Boolean Prime Ideal Theorem?

If there is a bijection $\varphi:x\to y$ between two sets $x$ and $y$, we use the notation $x\simeq y$. The Weak Power Hypothesis is the following statement: (WPH) For all sets $x, y$, whenever ${\...
2 votes
1 answer
64 views

Infinite decreasing sequence for class relation without minimal elements

Let us assume $<$ is some class relation without minimal elements, meaning $\forall a, \exists b, b< a$. This means that for every $n\in\omega$, one can build a decreasing function $f$ with ...
6 votes
2 answers
441 views

Can Deep Choice entail Axiom of Choice?

Deep Choice: $\forall X \ [\forall a,b \in X \, ( a \neq \emptyset \land (a \neq b \to a \cap b = \emptyset)) \to \\ \exists Y \exists f \,(f: X \rightarrowtail Y \land \forall x \in X \,( f(x) \in \...
1 vote
0 answers
124 views

How much choice we can get from this partition principle?

For every set $X$ there cannot be a partition on $X$ of a larger size than the set $\iota``X$ of all singleton subsets of $X$. Formally: $$\begin{align} \forall X \forall P: P \text { is a partition ...
4 votes
1 answer
799 views

About the relationship between the generalized continuum hypothesis and the axiom of choice

I was trying to get a short, intuitive proof of Sierpinski’s theorem (gch implies axiom of choice) and I could but only by using the following gch2 for the generalized continuum hypothesis gch. gch: ...
13 votes
3 answers
747 views

How to make countably closed forcing "nice" without choice

When working over a model $V$ of $ZFC$, countably closed forcings are extremely nice: If $\mathbb{P}$ is countably closed, then $V[G]$ has no new $\omega$-sequences of elements of $V$. In ...
16 votes
1 answer
1k views

Does Urysohn's Lemma imply Dependent Choice?

It's widely known$^{1}$ that in the proof of Urysohn's Lemma (UL) one uses the Principle of Dependent Choice (DC). Inspired by the equivalence between DC and Baire's Category Theorem$^{2}$, I'd like ...
41 votes
2 answers
2k views

On the difference between two concepts of even cardinalities: Is there a model of ZF set theory in which every infinite set can be split into pairs, but not every infinite set can be cut in half?

An interesting question has arisen over at this math.stackexchange question about two concepts of even in the context of infinite cardinalities, which are equivalent under the axiom of choice, but ...
0 votes
0 answers
49 views

Does parallelism of cardinal comparison between sets and their power sets, enact a form of choice? [duplicate]

Let $ * $ range over cardinal relations $ \{<,<>\}$; if we add the following axiom to $\sf ZF$, would that prove a known form of choice? Parallelism: $ |x| * |y| \leftrightarrow |\mathcal P(...
2 votes
0 answers
75 views

Is "strongly unbounded logics are unbounded" equivalent to "no descending sequence of cardinals"?

This question is motivated by Vaananen's paper Generalized quantifiers in models of set theory. Say that a (set-sized, regular) logic $\mathcal{L}$ is unbounded if there are $\mathcal{L}$-sentences $\...
65 votes
9 answers
13k views

Axiom of choice, Banach-Tarski and reality

The following is not a proper mathematical question but more of a metamathematical one. I hope it is nonetheless appropriate for this site. One of the non-obvious consequences of the axiom of choice ...
22 votes
8 answers
3k views

Simpler proofs using the axiom of choice [duplicate]

I am looking for examples of results which may be proven without resorting to the axiom of choice/Zorn lemma/transfinite induction but whose proof is quite simplified by the use of the axiom. For ...
4 votes
0 answers
242 views

Can we have full choice prior to Reinhardt cardinals?

Working in $\sf ZF + Reinhardt \ cardinal$, can we have full choice over all stages $V_{\alpha < \kappa}$ where $\kappa$ is the Reinhardt cardinal, i.e., the critical point of the elementary ...
1 vote
0 answers
65 views

Can this method let choiceless large cardinals be smaller than cardinals compatible with choice?

Recall question "Can we have this sequence where choice fails and returns?" Can that theory be extended with requiring the $\mathcal V_n$'s to fulfill a choiceless large cardinal extension ...
4 votes
1 answer
232 views

Can we have this sequence where choice fails and returns?

Can we have a sequence of transitive sets $\langle\mathcal V_0, \mathcal V_1, \mathcal V_2,...\rangle$, all modeling $\sf ZF$, such that $\mathcal P(V_n) \subset \mathcal V_{n+1}$, and the cardinality ...
49 votes
7 answers
7k views

Zorn's lemma: old friend or historical relic?

It is often said that instead of proving a great theorem a mathematician's fondest dream is to prove a great lemma. Something like Kőnig's tree lemma, or Yoneda's lemma, or really anything from this ...
8 votes
1 answer
1k views

Is there a form of choice that can elude Kunen's inconsistency theorem?

When it is said that Kunen inconsistency theorem proves that given $\sf ZFC$ no elementary embedding can exist from the universe to itself. Most references quote full choice in stating that result, ...
63 votes
14 answers
6k views

Unnecessary uses of the axiom of choice

What examples are there of habitual but unnecessary uses of the axiom of choice, in any area of mathematics except topology? I'm interested in standard proofs that use the axiom of choice, but where ...
9 votes
1 answer
3k views

Axiom of choice and non-measurable set

We know that existence of a Lebesgue non-measurable set follows from the Axiom Of Choice. Is the converse true? That is, does the existence of a Lebesgue non-measurable set imply the Axiom Of Choice?...
12 votes
2 answers
792 views

Ultrafilter lemma for arbitrary lattice

Can someone kindly confirm whether the ultrafilter lemma for arbitrary (i.e., not necessarily Boolean) bounded lattices is equivalent to Zorn's lemma? To be precise, if $\mathbf{L} = (L, \leq, \land, \...
9 votes
3 answers
1k views

Example of a topos that violates countable choice

At this nLab page we have the line In contrast, any topos that violates countable choice, of which there are plenty, must also violate internal COSHEP. It doesn't give an example, and neither does ...
12 votes
1 answer
421 views

Is there a more modern account of the main results of "Adding Dependent Choice" by D. Pincus?

I would like to read Pincus' article Adding dependent choice, where he proves, among other things, the consistency of the theory $\mathsf{ZF+DC+O+\neg AC}$, where $\mathsf{DC}$ stands for Dependent ...
12 votes
0 answers
416 views

On the division paradox

This question is partly motivated by Timothy Chow's recent question on the division paradox. Say that a set $X$ admits a paradoxical partition if and only if there is an equivalence relation $\sim$ on ...
3 votes
0 answers
198 views

Weak Power Hypothesis with injections instead of bijections

Let $x,y$ be sets. We use the following notation: $x\simeq y$ means that there is a bijection $\varphi:x\to y$, and $x\leq y$ means that there is an injection $\iota:x\to y$. The Weak Power ...

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