/sci/ - Science & Math

>>11101924
>>11102115
>>11102253

No it doesn't. Set theory, like group theory, are just common examples for the kind of theories logic enables you to write down.
There is a notion of collection in logic, namely the universe of discourse, but it's not a "set". You don't speak of membership (or intersections, etc.) there.
Logic stands on its own and just requires the capacity of rewriting

https://en.m.wikipedia.org/wiki/Abstract_rewriting_system

>>11102019
No, you can write down category theory as a first order theory and define subobjects via what amounts to characteristic functions in pullbacks. This is just commuting diagrams

>>11057249
Again, think of trees.
E.g. take n^2 prime numbers p_0,p_1,...,p_n^2 and for each of your matrix entries m_0,m_1,...,m_n^2, consider the number that's the product over all n^2 j's of pj^mj.
Now you embedded all your matrices into N

If you believe in LEM, you can now use

https://en.m.wikipedia.org/wiki/Schr%C3%B6der%E2%80%93Bernstein_theorem

to argue for le existence of a bijection.

But also you could use ordinals and that you can get arbitrarily deep with arbitrary long "rows".
The only thing you can't do is going to infinite objects in their own, like the elements of N->{0,1} which each is an infinite choice in its own