All Questions
Tagged with axiom-of-choice axioms
9
questions
10
votes
0
answers
212
views
Is any choice axiom other than WISC inherited by Grothendieck topoi?
It is well known that even if one works with say ZFC as a base theory, Grothendieck topoi do not in general satisfy even fairly weak axioms like countable choice or small violations of choice and one ...
- 1,843
11
votes
1
answer
518
views
Is every set being cardinal definable consistent with ZF + negation of Choice?
Recall the definition of cardinal definable, where every set being cardinal definable is proved consistent relative to ZF + V=HOD. To re-iterate it:
$Define: X \text { is cardinal definable} \iff \\\...
- 10.2k
10
votes
1
answer
312
views
Is choice over definable sets equivalent to AC over axioms of ZF-Reg.?
If we add the following axiom schema to ZF-Reg., would the resulting theory prove $\sf AC$?
Definable sets Choice: if $\phi$ is a formula in which only the symbol $``y"$ occurs free, then:
$$\forall X ...
- 10.2k
3
votes
2
answers
526
views
Relation between AC and the axiom of foundation
The fact that the axiom of foundation doesn't imply the axiom of choice is pretty standard (the model Cohen created to prove the consistency of $\neg AC$ models the axiom of foundation as well), and ...
- 14.4k
4
votes
1
answer
168
views
Does this axiom (a weak form of class valued choice) has a name?
At some point in my work (which has nothing to do with set theoretics foundation) I need to consider the following axiom:
For any set $X$, any class $V$ with a surjective map $f : V \...
- 40.8k
7
votes
8
answers
1k
views
Result that follows from ZFC and not ZF but are strictly weaker than choice
A number of results that people use that require the axiom of choice (i.e. do not follow from ZF alone) are known to actually imply the axiom of choice. Therefore, one might naturally wonder whether ...
- 15.5k
12
votes
2
answers
7k
views
Is it possible to show that an infinite set has a countable (infinite) subset, without using the Axiom of Choice?
Let X be an infinite set.
Is it possible to show the existence of a countably infinite subset of X without using the Axiom of Choice?
- 185
28
votes
11
answers
7k
views
Does the Axiom of Choice (or any other "optional" set theory axiom) have real-world consequences? [closed]
Or another way to put it: Could the axiom of choice, or any other set-theoretic axiom/formulation which we normally think of as undecidable, be somehow empirically testable? If you have a particular ...
- 2,383
6
votes
2
answers
2k
views
Axiom of Computable Choice versus Axiom of Choice
What would be the consequence of requiring that any choice function be computable; i.e. using as the foundational basis ZF + ACC? Does it make a difference if we admit definable functions?
I guess I ...
- 975