# 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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The Division Paradox is the fact that there are models of ${\sf ZF \neg C}$ in which a set can be partitioned into a set that is bigger than it — equivalently, in which there are sets $X$ and $Y$ such ...
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### 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 ...
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### 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 ...
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### 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  a ...
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### 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 ...
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### 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 ...
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### 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 $\iotaX$ of all singleton subsets of $X$. Formally: \begin{align} \forall X \forall P: P \text { is a partition ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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, ...
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### 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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### Partitioning a set of cardinality $\kappa$ into more than $\kappa$ disjoint subsets

There is an old result due to Mycielski and Sierpiński, and popularized in a Monthly article by Taylor and Wagon (A Paradox Arising from the Elimination of a Paradox; see also this MO answer), that ...
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### 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 ...
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### Compatibility of $\mathsf{SVC}$ and Reinhardtness

Can we prove the consistency of $\mathsf{ZF+SVC}$ + "There is a Reinhardt cardinal?" (Preferably from the consistency of $\mathsf{ZF}$ with a Reinhardt cardinal, but using a stronger ...
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### Automorphisms of projective spaces, and the Axiom of Choice

It is known that upon not accepting the Axiom of Choice (AC), there exist models of ZF in which there are projective spaces (over a division ring) with a trivial automorphism group. (This is a truly ...
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### What can be preserved in mathematics if all constructions are carried out in ZF?

This is inspired by this discussion. I see that the debates about the necessity of the axiom of choice in this or that statement are still ongoing. In this regard, I became interested in whether there ...
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### In characterizing finiteness via $\mathsf{AC}_\omega$, do we need to use large sets?

The usual proof that "Dedekind finite = finite" from $\mathsf{ZF+AC_\omega}$ goes by, given a Dedekind-infinite set $X$, applying countable choice to the sequence $(X^n)_{n\in\omega}$. It ...
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### A Krull-like Theorem and its possible equivalence to AC

A well known equivalent of the Axiom of Choice is Krull's Maximal Ideal Theorem (1929): if $I$ is a proper ideal of a ring $R$ (with unity), then $R$ has a maximal ideal containing $I$. The proof is ...
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### Is any choice axiom other than WISC inherited by Grothendieck topoi?

It is well known that even if one works with say ZFC as a base theory, Grothendieck topoi do not in general satisfy even fairly weak axioms like countable choice or small violations of choice and one ...
This question is an attempt to chisel away at this earlier question of mine, which in retrospect may be rather intractable. Throughout, we work in $\mathsf{ZF}$. Briefly (see the linked question for ...