[Math] Book on the Rigorous Foundations of Mathematics- Logic and Set Theory

logicreference-requestself-learningset-theory

I am asking for a book that develops the foundations of mathematics, up to the basic analysis (functions, real numbers etc.) in a very rigorous way, similar to Hilbert's program. Having read this question: " Where to begin with foundations of mathematics"
I understand that this book must have:

  • Propositional Logic
  • First Order-Predicate Logic
  • Set Theory

Logic however must not depend on set theory! I have tried reading many pdf notes on the first two but have been dissapointed by the usage of notions and concepts from Set Theory.

So what do I want? A book that builds up these $3$ from ground $0$ and develops the foundations of mathematics up to the Axioms of ZFC and simple consequences like the existence of the real number field. As such, it is not neccessary for this book to contain the incompleteness theorems, cardinality etc. It must however be rigorous and formal in the sense that when I finish it, I have no doubt that the foundations are "solid".

Final notes: It would be preferable if it were made for self study (but that's not neccessary). You can also suggest up to 3 books that discuss the topics above, beware however as circular definitions must be avoided. Rigor in other words, is the most important thing I am asking for.

PS: There have been other questions here on the foundations of logic as this one. They do not answer my question however, as rigor is not (over)emphasised. I believe this is not a duplicate and I hope you see that as well.

Thank you in advance

Best Answer

Gosh. I wonder if those recommending Bourbaki have actually ploughed through the volume on set theory, for example. For a sceptical assessment, see the distinguished set theorist Adrian Mathias's very incisive talk https://www.dpmms.cam.ac.uk/~ardm/bourbaki.pdf

Bourbaki really isn't a good source on logical foundations. Indeed, elsewhere, Mathias quotes from an interview with Pierre Cartier (an associate of the Bourbaki group) which reports him as admitting

'Bourbaki never seriously considered logic. Dieudonné himself was very vocal against logic'

-- Dieudonné being very much the main scribe for Bourbaki. And Leo Corry and others have pointed out that Bourbaki in their later volumes don't use the (in fact too weak) system they so laboriously set out in their Volume I.

Amusingly, Mathias has computed that (in the later editions of Bourbaki) the term in the official primitive notation defining the number 1 will have

2409875496393137472149767527877436912979508338752092897

symbols. It is indeed a nice question what possible cognitive gains in "security of foundations" in e.g. our belief that 1 + 1 = 2 can be gained by defining numbers in such a system!

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