Questions tagged [axiom-of-choice]
The axiom of choice is a common set-theoretic axiom with many equivalents and consequences. This tag is for questions on where we use it in certain proofs, and how things would work without the assumption of this axiom. Use this tag in tandem with (set-theory).
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Is every (infinite) permutation the composition of 2 involutions in ZF?
It is well known that any permutation on a finite set is the product of two involutions. I've wondered about what can happen in infinite sets.
Asuming the axiom of choice, every permutation is still a ...
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Discontinuous linear map and AC
The question arises when I am constructing an elementary proof for the following claim:
Given a normed vector space $V$, the following are equivalent:
$V$ is finite dimensional
Every linear map $T:V\...
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Axiom of Choice in characterizing openness in subspace
Below is the typical characterization of open sets in a subspace $Y$ of a metric space $X$.
$E$ is $Y$-open iff there exists an $X$-open $S$ such that $E = S \cap Y$.
The forwards direction usually ...
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Why is the Axiom of Choice Necessary in ZFC
Within the framework of Zermelo-Fraenkel set theory with the Axiom of Choice $(ZFC)$, when we considered the method of constructing the set of natural numbers, we regarded it as the smallest inductive ...
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Are the sets of the form $a=\{a\}$ different?
Suppose we adopt all the ZF axioms except the axiom of foundation. I suppose even adding the AC would not harm my question.
Somewhere on the internet, I read that in this case, the axiom of ...
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Truncated Tarski's theorem without axiom of choice
I read there that the fact about equivalence of $A$ and $A^2$ for any infinite set $A$ and the axiom of choice are equivalent. But what if we prove it only for sets that have cardinality of $\aleph_n,...
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uniform well-ordering and constructibility
When comparing between V=L and AC, one of the things that gets my attention is that, if we switch to an external perspective and don't care about first-order expressibility, in models of V=L we have a ...
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Is ACC implicitely involved in a construction?
Suppose that $X$ is a (edit: separable) complete metric space and $\{X_j:j\ge 1\}$ is a partition of $X$ (each $X_j\ne \emptyset$). Given a (continuous) function $f:X\to \mathbb{R}$, I construct
$$
h(...
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The relation between the cardinality of Bore $\sigma$-algebra and axiom of choice
Under axiom of choice, the cardinality of Borel $\sigma$-algebra $B$ is $\mathfrak{c}$.
In this proof axiom of choice is used three times: To prove
$\omega_1$-times recursion is sufficient,
each $|B_{...
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Construction of Proof: Zorn's lemma implies Axiom of choice
I have come across the prove that [Zorn's Lemma ==> AC] but am confused about the central statement, namely that we can take a set of all choice functions on subsets of X (lets just call it X, I ...
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Cantor-Bendixson theorem and AC
For context, the Cantor-Bendixson theorem states that a closed subset $A$ of a Polish space can be written as the union of a perfect subset and a countable set $A=P\cup C$.
Now, I know two proofs of ...
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Why can't $\mathbb{R}/\mathbb{Q}$ be linearly-ordered without Axiom of Choice?
This Question has an answer which is the only source that I can find about how $\mathbb{R}/\mathbb{Q}$ cannot be linearly ordered. I couldn't manage to open either of the source links provided in the ...
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Is the Axiom of Dependent Choice necessary in this proof?
While typically, the Axiom of Choice and its peripheral arguments are not emphasized in one's first exposure to Real Analysis, I am trying to be as rigorous as possible in my learning as an axiomatic ...
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Use of (weak forms of) AC for elementary embeddings proof
I encounter this issue when going through equivalent characterizations of measurable cardinals. For completeness, let me reproduce the statement:
For ordinal $\kappa$, the following are equivalent.
...
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Is the existence of a discontinuous linear map from a Hilbert space equivalent to the Axiom of Choice?
All constructions of a discontinuous linear map from an Hilbert Space, that i have seen, rely on using the Axiom of Choice. A lot of theorems that heavily rely on AC are equivalent to AC itself so i ...
