Where to begin with foundations of mathematics

I would like to know more about the foundations of mathematics, but I can’t really figure out where it all starts. If I look in a book on axiomatic set theory, then it seems to be assumed that one already have learned about languages. If I look in a book about logic and structure, it seems that it is assumed that one has already learned about set theory. And some books seem to assume a philosophical background. So where does it all start?

Where should I start if I really wanted to go back to the beginning?

Is it possible to make a bullet point list with where one start? For example:

  • Logic
  • Language
  • Set theory

EDIT: I should have said that I was not necessarily looking for a soft or naive introduction to logic or set theory. What I am wondering is, where it starts. So for example, it seems like predicate logic comes before set theory. Is it even possible to say that something comes first?

Answer

There are different ways to build a foundation for mathematics, but I think the closest to being the current “standard” is:

When rigorously followed (e.g., in a Hilbert system), classical logic does not depend on set theory in any way (rather, it’s the other way around), and I believe the only use of languages in low-level theories is to prove things about the theories (e.g., the deduction theorem) rather than in the theories. (While proving such metatheorems can make your work easier afterwards, it is not strictly necessary.)

Attribution
Source : Link , Question Author : Thomas , Answer Author : jwodder

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