/sci/ - Science & Math

>>10093115
ad:
* maybe now it also makes sense why the axiom of choice is equivalent to the well ordering theorem, and zorns lemma, and Hausdorff principle
>Suppose a partially ordered set P has the property that every chain in P has an upper bound in P. Then the set P contains at least one maximal element.
>In mathematics, the well-ordering theorem states that every set can be well-ordered.
>The Hausdorff maximal principle ... states that in any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.
It's all cheat codes to make sets have some choosing structure even if you actually don't know.

* also worth pointing out that postulating some material set theory - e.g. where you force people to build everything from the empty set and then some super-sets, this gives you already a lot of necessary properties. Zermelo Freankel actually in itself wont talk about sets of fruits or humans, just about sets or sets of sets (with not infinite decending ones). This sort of boundary condition e.g. makes it possible that if I give you a finite set of things of arbitrary size, then the corresponding choice function can be shown to actually exist in this context. But it's not the case for infinite sets of sets.

Pic related is a weaker form worth considering

>>10092610
It's independent of Zermelo-Freankel set theory, not more or less.

>>10091914
No, you're mistaken.
Even if you're given a set of sets where you know that all sets involved are of infinite cardinality, then without the axiom of choice, it's still not a given that you construct a function.
His elaboration actually cuts quite straight to the point.

>>10091813
>Could someone explain a situation in which the AC is unintuitive or perhaps incorrect?
Yes, but you need to take a step back and reconsider what stance you want to take on provability resp. "constructibility" of mathematical objects. So here's the situation you look for:

Let's assume [math] X = {a_1, a_2, a_3, ...} [/math] is an infinite set of sets where each element, [math] a_{43} [/math] say, has seven elements, i.e. for all k we have that [math] |a_k| = 7 [math].
Okay, now I give you the following task: State a function [math] f [math] with [math] X [math] as domain, which for each [math] a_k [math], returns an element.

..
..
..
Got a function yet?
No, because I didn't tell you anything about the [math] a_k [/math] other than they they contain something.

Now if I were to say that the element of each [math] a_k [/math] have a strict order attached to them, you could actually go on. E.g. if I were to tell you all [math] a_k [/math] hold natural numbers, or if I were to tell you the [math] a_k [/math] hold tuple of fruits and no two tuples of [math] a_k [/math] have the same size. If you know anything about the [math] a_k [/math], you have a chance of going on. If I tell you all [math] a_k [/math] are orderable finite sets, then
>choose the smallest element from each [math] a_k [/math]
is one way to construct a choice function.
But in general, you can make up a context where you have nothing to work with.

The axiom of choice says "well, we're just gonna axiom you can say 'muh f' and go on with your proof."
A constructivist would say here's where you need to stop due to lack of info.