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# Tagged Questions

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 ...

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

Is the following consistent with $\text{ZF}$? There exists a set $S=\{x_1,x_2,x_3,...\}$ such that: $|x_{i+1}| < |x_i|$ $\forall m,n \in S (|P(m)|=|P(n)|)$ Where cardinality $||"$ is ...
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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?

The question of existence of sets $x,y$ such that $$|x|<|y| \wedge |P(x)|=|P(y)|$$ is known to be independent of $\text{ZFC}$! But are there known examples of sets fulfilling the above condition ...
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### When does $A^A=2^A$ without the axiom of choice?

Assuming the axiom of choice the following argument is simple, for infinite $A$ it holds: $$2\lt A\leq2^A\implies 2^A\leq A^A\leq 2^{A\times A}=2^A.$$ However without the axiom of choice this doesn't ...
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### Do choice principles in all generic extensions imply AC in $V$?

It's well-known that not all choice principles are preserved under forcing, e.g. in this answer https://mathoverflow.net/a/77002/109573 Asaf shows the ordering principle can hold in $V$ and fail in a ...
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### The Tall Tale of Terminating Transfinite Towers

The transfinite tower of iterative automorphisms of a group $G$ is simply definied to be the following chain of the groups where $G_{\alpha+1}=Aut(G_{\alpha})$ for each ordinal $\alpha$ and the direct ...
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### Why worry about the axiom of choice?

As I understand it, it has been proven that the axiom of choice is independent of the other axioms of set theory. Yet I still see people fuss about whether or not theorem X depends on it, and I don't ...
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### The partial preorder on $\mathbb N$ generated by the finite axioms of choice

Let $\mathsf C_n$ denotes the statement: for any family $\mathcal F$ of $n$-element sets there exists a choice function (i.e., a function $f:\mathcal F\to\bigcup\mathcal F$ such that $f(F)\in F$ for ...
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### Graphs without maximal vertex-transivite subgraphs

The axiom of choice is of no use when trying to prove that every vertex-transitive subgraph is contained in a maximal vertex-transitive subgraph, because a union of an ascending chain of vertex-...
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### Relations of axioms of choice

We start with $ZF$. The axiom of countable choice, $AC_\omega$, says that any set product of nonempty sets with a countable index set is nonempty. For any $ZF$-definable set $A$, we should be able ...
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### Are classes still “larger” than sets without the axiom of choice?

Classes are often informally thought of as being "larger" than sets. Usually, the notion of "larger" is formalized via an injection: $B$ is "at least as large" as $A$ iff there is an injection from $A$...
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### Can There be a 1 dimensional Banach-Tarski paradox in the absence of choice

Let $\mathbb{R}$ act on itself by translation. Then there is no finite decomposition of a unit interval into pieces which, when translated, yields two distinct unit intervals. More formally does ...
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### Measurable functions with non measurable image

I am just curious about examples of measurable functions $f:[0,1]\to[0,1]$ such that $f[0,1]$ is not measurable. This is motivated by the question Is measure preserving function almost surjective?, ...
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### Does the existence of a unique chromatic (possibly transfinite) number for every (possibly non-finite) simple graph imply the axiom of choice?

Assuming the axiom of choice I can write for any cardinal number $\kappa$ and any simple graph $G$ that a function $f$ is a $\kappa\text{-coloring}$ of $G$ if and only if the cardinality of the image ...
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### In what ways is ZF (without Choice) “somewhat constructive”

Let me summarize what I think I understand about constructivism: "Constructive mathematics" is generally understood to mean a variety of theories formulated in intuitionist logic (i.e., not assuming ...
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### Does the axiom of choice follow from the statement “Every simple undirected graph is either connected, or its complement is connected”?

Using the Well-Ordering Principle, which is equivalent to the Axiom of Choice, it can be proved that (S): for every simple, undirected graph $G$, finite or infinite, either $G$ or its ...
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### Can we prove the epsilon theorems without the axiom of choice?

Hilbert's epsilon $\epsilon$ is a quantifier. It follows the rule that if $\exists x. p(x)$, for some predicate $p$, we can infer $p(\epsilon x. p(x))$. Semantically, it represents picking some ...
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### Forcing over models without the axiom of choice

In the vast majority of papers forcing is always developed over ZFC. Not surprisingly too, since infintary combinatorial principles are often used to prove results based on properties such as chain ...
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### A model of ZF without a well-ordering of the reals in which any two sets of reals are comparable

Let us say that two sets $A$ and $B$ are comparable if there is an injection from $A$ to $B$ or there is an injection from $B$ to $A$. Obviously, in a model of ${\rm ZFC}$ any two sets are comparable ...
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### Chromatic number of a topological space

Here is a question I asked myself years ago. Since it is not really in my field, I hope to find some (partial) answers here... Since it was unclear, I precise that I am looking for an answer in ZFC, ...
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### Is the non-triviality of the algebraic dual of an infinite-dimensional vector space equivalent to the axiom of choice?

If $V$ is given to be a vector space that is not finite-dimensional, it doesn't seem to be possible to exhibit an explicit non-zero linear functional on $V$ without further information about $V$. The ...
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### Undetermined games of “overdetermined” type

This is motivated by a previous question of mine, but I think it is ultimately more interesting (and hopefully easier to answer in the positive). In that question, a class of games (on $\omega$, of ...
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### The axiom of choice as a consequence of a stronger semantics?

I've never had a problem with the axiom of choice, but it has often confused me how many authors find full choice so much different from finite choice. In my head they seem quite similar. We are ...
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### Is it known whether every $\omega$-tree with an infinite antichain has an infinite chain in $\mathsf{ZF}$?

In this paper by Good and Tree, the following result is mentioned without proof as part of Proposition 6.5: Each of the following statements imply those beneath it. The countable union of ...
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### Unique Existence and the Axiom of Choice

The axiom of choice states that arbitrary products of nonempty sets are nonempty. Clearly, we only need the axiom of choice to show the non-emptiness of the product if there are infinitely many ...
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### Hahn-Banach and the “Axiom of Probabilistic Choice”

Stipulate that the Axiom of Probabilistic Choice (APC) says that for every collection $\{ A_i : i \in I \}$ of non-empty sets, there is a function on $I$ that assigns to $i$ a finitely-additive ...
### Relation between well-orderings of $\mathbb{R}$, and bases over $\mathbb{Q}$
The following question arose from a discussion about the definability of bases of $\mathbb{R}$ as a $\mathbb{Q}$-vector space. (ZF without AC) something we can note is that the existence of a (...
I'm not sure this question is more suitable for MO or for MSE, so feel free to move it to MSE if necessary. I work here in ZF theory. Consider the following statements: $(C)$ Axiom of choice: for ...