Questions tagged [axiom-of-choice]

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. (from MathOverflow)

Filter by
Sorted by
Tagged with
4 votes
1 answer

How important is global choice (a la Lean, HOL Light, Isabelle/HOL) practically?

Choice is indispensable for much of modern classical mathematics. Therefore, most proof assistants offer it as part of their standard library. The most powerful version is sometimes called global ...
  • 4,690
14 votes
3 answers

Do you need a Hilbert style Epsilon operator for definitions in set theory?

I've started to play with mechanizing some set theory stuff. I'm not sure if I want a constructive flavor or not yet. Anyhow you can do stuff like axiomize the empty set $$ \top \vdash \exists P. \...
9 votes
1 answer

What axioms do I need to search the naturals?

Theorem search {P : nat -> Prop} (dec : forall n, {P n} + {~P n}) : ~~(exists n, P n) -> {n | P n}. Admitted. I don't think this is provable in Coq without ...
  • 368
16 votes
3 answers

Well-foundedness: classical equivalence of no infinite descent and accessibility

I have often seen the claim that in a classical setting, well-foundedness of a relation > defined as the absence of an infinite descent ...