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My goal right now is to gain a deep understanding of how to talk about mathematical objects formally. The presentation of how to do this in most books is generally to "assume some basic set theory," and then go about developing logic, or "begin with basic notions from logic," and develop set theory. I'm wondering if there is a blended approach, which would first, perhaps, define a little set theory, then a little logic, then a little more set theory, etc., and eventually you'd have enough to go on where it wouldn't be so hard just build out the topics separately.

I know that this is going to be a less direct, and probably far uglier way, of going about the development of the very foundations of mathematics, but it is intellectually satisfying to me to know that it has been done. I also believe it appeals to the general mathematical ideal of rigor. The qualities of such a presentation I'm looking for is:

  • Use the fewest "basic notions"/assumptions as possible
  • Only build off notions that have not been assumed or derived (e.g. do not invoke the general "basic notions of set theory" "basic logical notions" unless those notions have been clearly laid out on the table)

Thank you!

Axel
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  • The set theory axioms are written in terms of logic. It sounds like you just want a text on basic logic that doesn't appeal to set theory? – Mark S. Jun 21 '20 at 17:29
  • You might be interested in Kunen's exposition of Formalism in his book "Set Theory." It is the very first chapter and the first section "On doing things twice" sounds very similar to what you describe above. – Jonathan Schilhan Jun 21 '20 at 22:21
  • Thank you @Jonathan, I will look into that! – Axel Jul 21 '20 at 01:45
  • @MarkS., yes, or the other way around. Or, simply, a text that names all the assumptions being used, even if they are "naively" stated. I have been reading Moschavakis' Notes On Set Theory and he seems to do this to some extent. I will have to keep reading to see where it goes though. – Axel Jul 21 '20 at 01:45
  • I feel like most introductions to logic don't appeal to set theory. But I'm glad you've found notes that seem to do something like what you were looking for. – Mark S. Jul 21 '20 at 10:28

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