/sci/ - Science & Math

>>14825792
If an uncountable family of non-empty sets C={X,Y,Z,...} has only sets that are non-empty because for every X you know of an x in X, then indeed you already got a choice function and the statement is identically true.
If the uncountable family of non-empty sets has only non-empty set "because reasons", then the axiom of choice exactly postulates that there is a selection. (There just is OKAY?)

In lattice theory, a well-ordering is a relation < between U and V which in set theory is modeled as some subset of of UxV.
Now it's consistent in ZF that there exists no well-ordering on the reals. ZF can't prove there's a subset RxR which is a well-ordering relation.
Now you may make your theory stronger and adopt the existence postulate AoC. You now got the theory ZFC and what happens is that in this theory it's provable that "there exists" a well-ordering of the reals.
That is, ZFC proves that there is a subset of RxR which puts all elements of R in some order (one way or the other)
That's nice, right - did we win something? No, it turns out that even in ZFC, it's still consistent that there is no definable well-ordering of the reals. That is to say, we can't possibly write down a fixed well-ordering in ZFC alone, in that we could take ANY two reals and that fixed well-ordering and it tells you how the two are ordered.
AoC simply says that something exists but it doesn't give you more or less than that. You get existence statements which make the ZFC in turn sound like they are just theorems about finite sets, where things hold as you expect. Adopting ZFC makes the theory have formally less surprises, but the existence statement are hollow.
Another example: ZF does not prove that all vector spaces have a basis. ZFC does prove it. But there's still vector spaces for which you provably can't write down the basis. ZFC theorems just say things like "all vector spaces have basis" and it sounds sensible, because it's modeled on things you know.

Hope that helps

>>12431918
Big fan of Curry-Howard, but Hegel is our last chance to meaning so we best not reject it.

>>11114224
A given spin (some rational number) is an indexing for a representation.
Learn (Lie group) representation theory.

https://en.wikipedia.org/wiki/Particle_physics_and_representation_theory

Also, just in case
https://youtu.be/zsLOVYTLt90