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For over 8 months, I have been actively looking for a resource that rigorously tackles mathematical logic - at the same level described in this lecture, timestamp 56:09. A resource which minds the proper order of definitions and uses special constructs such as "pre-mathematical numbers" to get around the problem of natural numbers/sets being undefined at the lowest levels (definitions level) of logic.

All of the mathematical logic books that I've evaluated so far mixes set theory in the logic definitions. But this is not the way that I prefer to study the subject. Could someone please refer me to a resource on logic that has mechanisms to get around circular definitions?

I am well-aware that set theory and logic are very intertwined, and I don't mean to pull them apart. I'm just interested in the proper organization of ideas similar to the standard mentioned here.

Thanks!

lolm2k
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    What exactly is a "pre-mathematical number"? – Noah Schweber Apr 28 '18 at 16:41
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    It's in the lecture link. I could explain it here, but he explains it better there w/ context. – lolm2k Apr 29 '18 at 00:54
  • Just to talk about formal languages, you need some basic notions that you might call "pre-mathematical". However, there is no circularity, because you only need those notions in the metalanguage you are using to talk about the formal languages. In the lecture you have been studying, there must already have been an appeal to notions like the sequences of symbols that make up a proposition, so it is a bit odd that the lecturer only raises this issue when he needs to talk about finite sets of propositions. His definition of an axiomatic system is also much weaker than is usual. – Rob Arthan Apr 30 '18 at 21:44
  • See George Tourlakis, Lectures in Logic and Set Theory. Volume 1: Mathematical Logic, Cambridge (2003) and specifically page 10. – Mauro ALLEGRANZA May 02 '18 at 11:33

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