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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