Set theory and category theory

[tgt]

They seem to be different fields but both try to underpin maths. There has been suggestions that set theory is problematic, where some paradoxes cannot be resolved. But how about Category theory? Any problems or paradoxes? Is it more promising then set theory?

 

[granpa]

paradoxes are a normal part of math and logic. they don't indicate that there is anything wrong. categories will also lead to paradoxes.

 

[tgt]

How successful will category theory be as a foundations of maths? I can't believe it doesn't use any sets?

 

[granpa]

I'm not an expert. I'm just as confused as you are about that.

 

[Tac-Tics]

How successful will category theory be as a foundations of maths? I can't believe it doesn't use any sets?

It depends what you mean by "successful". Will it ever be as powerful as our set theories? Answer: yes, it already is. Will it ever be accepted socially as a superior alternative to set theory? Lawls, no way.

Really, category theory *does* have sets. It just call them (and everything else) by a different name. The analog for a set is essentially the object. Morphisms are analogous to functions. While functions map members of sets to member of other sets, morphisms map members of objects to other objects.

The big difference is that CT almost *never* talks about the "members of an object". They don't value the members. Instead, they focus their energy on finding relations between morphisms, especially when chained together through function composition.

 

[Hurkyl]

Will it ever be accepted socially as a superior alternative to set theory? Lawls, no way.

My view of the future isn't quite so pessimistic.

Really, category theory *does* have sets. ... morphisms map members of objects to other objects.

A set is a member of a set-theoretic universe. Your statement is only true if we're studying some set-theoretic universe and the category happens to be a subcategory of it.

Yes, it's true that category theory has analogs of the idea of elements, and in some situations, the analogy can be effectively equivalent (e.g. in the Set, one of the categorical notions of 'element' of X is a function {{}}--->X). But the set-theoretic approach of trying to describe everything as a 'set with structure' is often misleading, awkward, or simply doesn't work.

 

[dodo]

But the set-theoretic approach of trying to describe everything as a 'set with structure' is often misleading, awkward, or simply doesn't work.

For the unillustrated :) , could you show an example? In which realms category theory do provide a better start point than set theory or other tradition?

 

[Hurkyl]

Wikipedia's article on the origins of topos theory is an interesting read.


Wikipedia also gives two examples of categories whose objects can't be viewed as sets with structure.

I should state a caveat, though -- if you add a 'large cardinal axiom' to your set theory, then while hTop cannot be viewed as 'small sets' with structure, you can represent them as 'large sets' with structure. However, the general recipe for such a construction is wholly unenlightening -- it's simply a restatement of the category structure. (It's directly analogous to the fact that any group can act on itself)