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I have been navigating in this "foundational world" of mathematics for a while now ( but certainly not long enough and not deep enough ) and have read a bit about many different topics : set theory, logic, Godel completeness and incompleteness theorems , model theory and proof theory.

I was wondering if anyone has a recommendation of a resource ( site, article, notes but mainly a book ) that gives a shallow overview of this foundational area of mathematics (Mathematical Logic) in a way to relate and show the connection between those many of the subjects i mentioned, kind of giving an introduction to each area.

I find that books of a certain topic are too specific ( and with the right reason ) and end up not providing the at least "basic" big picture of Foundational Mathematics I wish to have before delving deeper in my studies.

Thanks a lot!

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    $\begingroup$ I don't know about a specific book. But I think that any book covering the basics of first-order logic, and a book covering the basics of set theory, and then some time to connect the missing dots on your own should suffice. $\endgroup$
    – Asaf Karagila
    Commented Apr 27, 2014 at 15:01
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    $\begingroup$ I suggest you two recent books : Pavel Pudlak, Logical Foundations of Mathematics and Computational Complexity (2013) and Kenneth Kunen, The Foundations of Mathematics (2009). $\endgroup$ Commented Apr 27, 2014 at 17:08
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    $\begingroup$ I suggest also the books referenced by Peter Smith in this post for the historical background. $\endgroup$ Commented Apr 27, 2014 at 17:09
  • $\begingroup$ Thanks for those suggestions, you two :) $\endgroup$
    – nerdy
    Commented Apr 27, 2014 at 19:43
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    $\begingroup$ One book that I've found very useful and which introduced me to this subject is "Foundations of Set Theory" by Fraenkel, Bar-Hillel and Levy. The book is not too specific and introduces different systems. $\endgroup$ Commented Apr 28, 2014 at 2:49

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"Introduction to Metamathematics" and the even more basic "Mathematical Logic" by Stephen Cole Kleene both give a very good overview of the foundational area of mathematics and provided the basic big picture for me.

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