How are sets defined in mathematics?

[phoenixthoth]

first one seems ridiculously basic...the other one less so.

1. What is a set?

If I ask what is a vector, I can say it is something in a vector space. If I ask what is a group, I can see if it meets some simple criteria. The axioms of set theory say certain things are sets but is there something that could be turned into a definition for a set of the form, "x is a set iff... ." For groups G, there is a way to finish that line "G is a group iff... ." Same for vectors. What about sets? (Please no circular definitions like a set is a collection. I'm after the actual definition from math.)


2. Is there a way to turn the question of axiom independence into any other type of problem? I don't know much about it but does forcing do that? There must be some way to recontextualize axioms as generators of a group or something with the rules of deductive calculus being the group operations, or something, isn't there? Not saying it would be a group but some algebraic structure...

 

[matt grime]

x is a set (in some model) if and only if it satisfies the axioms of being a set. Some thing is not a vector until you specify a vector space and then verify it is an element of that space. Precisely the same holds for sets.

 

[Hurkyl]

The algebra of a set is the one with no function symbols. And behold: sets are in one-to-one correspondence with set-theoretic models of this algebra.

The theory of a set is the one with no function or relation symbols. And behold: sets are in one-to-one correspondence with set-theoretic models of this theory.

Surely this was not helpful, though! I agree with matt -- you first need to define a set theory, and then your sets are the 'elements' of that.


It turns out that bounded Zermelo set theories correspond with well-pointed topoi; it's even a 1-1 correspondence if you make add an additional axiom that's a consequence of replacement. Maybe an elementary version of the well-pointed topos axioms will be more to your liking?


Forcing can be used to prove axiom independence; I think it's basic idea is to adjoin an element satisfying the properties you want (e.g. N < X < PN), and then construct the set-theory generated by your original set-theory and this new element.