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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Countable Choice from Finite Sets
Consider the following 4 statements:
Axiom of countable choice
Axiom of countable choice from finite sets
Axiom of countable choice from Dedekind finite sets
Existence of a choice function for any ...
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A union of unions needn't be a union? (Sans AC) [duplicate]
For any collection $\mathscr C$ of sets, write $\Upsilon(\mathscr C)$ for the collection of arbitrary unions in $\mathscr C$. Now, I ask the innocent question of idempotency of $\Upsilon$:
Is $\...
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Does $A$-fold choice imply $|A| + |A| = |A|$ and $|A|\cdot |A| = |A|$?
Let $A$ be an infinite set. Then Zorn's lemma can be used to conclude that $A\times\{0, 1\}$, $A\times A$ and $A$ are all equinumerous (a proof is presented here). However, I am aware that for $A$ ...
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Is there a model of ZF not C where not every set of reals is Lebesgue measurable? [duplicate]
I know that there is a model of ZF set theory plus the negation of the axiom of choice where every set of reals is Lebesgue measurable. But is there also a model where not every set of reals is ...
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Principle in between BPI and AC
I have searched extensively in the literature, but all references I have consulted always place BPI (the Boolean Prime Ideal Theorem) as a sort of "cover" of the Axiom of Choice as far as ...
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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 ...
