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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In ZF, can a Dedekind-finite set (or even an amorphous set) be smaller than a non-trivial partition of its elements?
For context: A set is amorphous if every subset of it is either finite or cofinite but not both (in particular is infinite). A set is D(edekind)-finite if every injection into itself is a surjection. ...
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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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A weak version of König's Theorem - Does this depend on Choice?
Let $X_1,X_2,Y_1,Y_2$ be nonempty disjoint sets, where $|X_i|<|Y_i|$. Here, we take this to mean that there is an injection $X_i\to Y_i$, but there is no bijection $X_i\to Y_i$. My question is, is ...
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Assuming $(GCH)$ but not the axiom of choice, strongly inaccessible and weakly inaccessible coincide
My book says
"... If $GCH$ holds, then the notions of strongly inaccessible and weakly inaccessible cardinals coincide, ..."
In $ZFC$ I can prove this. But the paragraph from which I have excerpted ...
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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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How is the “ Axiom of choice is trivial in intuitionistic logic”?
In slide 28 of these slides, the author claims that the “Axiom of choice is trivial in intuitionistic logic” and that classical logic makes it a “ monster from outer space”. How is it trivial when it’...
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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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