What is system of logic/language used in study of mathematical logic/language?

[lolgarithms]

some broad questions about foundations of math.

what kind of logic/set theory should be used to study logic in mathematics/set theory?

I just have this question. It seems that insisting on using classical logic is an inherent bias in studying logics. How is it justified that set theory and other formalisms can be used in study of foundations of mathematics topics? (look at wikipedia articles, they have formal definitions of formal languages and stuff)

to pose this problem:
mathematics/logic is formalized using a system of formal symbols
this formal language is defined/axiomatized in semantics
these ideas have to be formalized, etc.

Also, is the reason that formal languages need other languages to describe them because of goedel's theorem:
that a language can't describe itself without making contradictory statements (IDK if this is true or not)
a sufficiently powerful axiomatization can't prove its own consistency
?

Sorry for this kind of broad topic. My ideas are also pretty foggy here

 

[honestrosewater]

You do not have to use classical logic to study logic. It sounds like you are worried about something tricky happening. If so, what? Your metalogical system isn't usually formalized itself or even made explicit. You use whichever logic is most useful.

The mathematical theories themselves have a logical component to them, so the theory that you are studying will determine the logic that you are studying. First-order logic happens to be the most prevalent, but it is not the only one studied or used.