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Results tagged with axiom-of-choice
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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.
14
votes
Infinitary logics and the axiom of choice
What you're basically describing is the result of replacing, in the usual definition of $\mathsf{ZF}$, schemes ranging over first-order formulas by schemes ranging over formulas in a different logic $ …
- 21.1k
18
votes
Accepted
Cantor-Bernstein with "weakly injective" functions
No, it is not provable in $\mathsf{ZF}$.
It is consistent with $\mathsf{ZF}$ that there is a sequence of disjoint two-element sets whose union is not countable, i.e. $\vert A_i\vert=2$ but there is no …
- 21.1k
3
votes
Can one show that the real field is not interpretable in the complex field without the axiom...
Embarrassingly, I actually made a correct version of this argument earlier on MSE; very belatedly, I've fixed things.
Here's a rather silly computability-based argument:
Suppose $\Phi$ were an interpr …
- 21.1k
2
votes
An equivalent of the axiom of choice?
With such shared/common knowledge problems, even when AC isn't involved, I think a crucial first step is to get away from the "story" version. For example, since this is only interesting once the set …
- 21.1k
42
votes
Zorn's lemma: old friend or historical relic?
I agree with the existing answers, but I personally like Zorn's lemma both pedagogically and mathematically for an additional reason: the "poset of partial solutions" that it introduces is a valuable …
- 21.1k
9
votes
Is Axiom of Choice equivalent to its version for families of sets, indexed by ordinals?
It is strictly weaker than choice. This is explained in Asaf Karagila's answer at MSE: the $L(\mathbb{R})$ of $L$ + $\aleph_1$-many Cohen generics witnesses this.
(There the principle is phrased for w …
CommunityBot
- 1
1
vote
Accepted
Is Proper Class Choice equivalent to Global Choice?
Yes, this is still equivalent: given any relation $R$, consider the new relation $$R^{bigrows}=\{\langle x,y\rangle: \exists a,b(y=\langle a,b\rangle\wedge \langle x,a\rangle\in R)\}.$$ Basically, $R^ …
- 21.1k
4
votes
Accepted
Does choice always hold in a model of ZF with point-wise parameter-free definable sets?
The following fleshes out the comments above by Asaf and Andreas.
First, note that the idea you outline at the end will not work: it implicitly assumes that the relation "$\varphi$ defines $a$" is de …
- 21.1k
1
vote
Axiom of Choice and Vitali's theorem
(I believe this material is in Jech's giant set theory book, but I don't have it in front of me right now.)
I am almost entirely certain that the existence of a non-measurable set of reals is not equ …
- 105
8
votes
Accepted
Amorphous proper classes in MK
Unless I'm missing something, the answer is no: we have a surjection $s$ from a given proper class to the class of ordinals - sending each element to its rank and then "collapsing" appropriately - and …
- 21.1k
4
votes
What are some kinds of models where DC holds?
Here's a negative answer to Q1 (it looks like the comment addressing this was flawed):
It is consistent with ZF that there is an infinite Borel (in fact, $F_{\sigma\delta}$) set $B$ with no greatest …
- 21.1k
4
votes
Relation between AC and the axiom of foundation
The following is an easy recipe for building a model of ZFA+AC with one atom, from a model $M$ of ZFC (it's trivial to modify this construction to include arbitrarily large sets of atoms, and even a p …
- 21.1k
3
votes
What are the minimal requirements for the definable hyperreal field plus transfer?
Since ACC is only applied to subsets of $\mathcal{P}(\mathbb{N})$ - that is, to families of sets of reals - the answer is that only the well-ordering of $\mathbb{R}$ is needed. Indeed, if $\mathbb{R}$ …
- 21.1k
5
votes
Accepted
The patterns of possibility for nontrivial automorphisms and nontrivial elementary embedding...
I think there's some confusion over automorphisms/embeddings which are "internal" vs. those which are "external." Some comments which I hope clear things up:
No model of ZF has nontrivial definable …
- 21.1k
14
votes
Independence of the countable axiom of choice
I'm going to play fast and loose with details here, but the outline is correct. To answer your new questions: no, there is no short proof. And this only shows one direction of the indpenedence of AC, …
- 21.1k