Are there any recent books or papers regarding (and focused on) this subject? I am very interested in learning as much as I can about undecidable theorems and methods/theory for testing if a math statement is undecidable. For context, I became interested by reading a passage in an old "Logic and Combinatorics" book from 1987 showing that in a certain formal system that you could not prove a modified finite ramsey theorem without referencing infinite sets. I'm not sure where to even place this as part of a subject when I search for it (and I have searched for it, most of the stuff I find is from a couple of decades ago). Is this considered purely part of logic or proof theory? Or is it just something that "pops up" when something you are trying to prove doesn't behave with your axioms?
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1generally, this falls within the realms of model theory. – Rushabh Mehta Aug 06 '21 at 00:47
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2The "stuff" you find from a couple of decades ago has not stopped being valid. To get as much learning as you can, start with the basics. Understand the definitions of formal logic and axiomatic theories developed in the first half of the last century before venturing beyond. – hardmath Aug 06 '21 at 01:00
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A great source for this is Harvey Friedman's extensive body of work. Check out his website, Harvey's Foundational Adventures.
There's also the excellent book by Stephen Simpson:
Stephen G. Simpson, Subsystems of second order arithmetic, Perspectives in Logic. Cambridge: Cambridge University Press; Urbana, IL: Association for Symbolic Logic (ASL) (ISBN 978-0-521-15014-9/pbk). xvi, 444 p. (2010). ZBL1205.03002.
For other material, a good search phrase is "reverse mathematics". The Wikipedia article Reverse Mathematics provides additional information and sources.
Mitchell Spector
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1Thank you so much this is amazing. I find it strange but I haven't really been as excited to learn about something since relativity in physics over a decade ago. Thank you very much. I'm off down the rabbit's hole. – James Shelton Aug 06 '21 at 05:06
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2(+1) This is a very nice entry-point answer, which seems exactly what the OP wanted. To your references (@James Shelton) I'd include the recent semi-technical exposition Reverse Mathematics. Proofs from the Inside Out by John Colin Stillwell (2018) -- See Notices AMS review and Amer. Math. Monthly review and Bull. Symbolic Logic review and (continued) – Dave L. Renfro Aug 06 '21 at 06:12
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1Math. Intelligencer review and Phil. Math. review and Math. Magazine review. Incidentally, the "Logic and Combinatorics" book mentioned (for others interested in specifics) is almost certainly Volume 65, edited by Stephen G. Simpson, of the AMS Contemporary Mathematics series, (continued) – Dave L. Renfro Aug 06 '21 at 06:12
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1which I got many years ago (but after 1987; probably around 1990) as a result of my interest in large numbers and large countable ordinals. – Dave L. Renfro Aug 06 '21 at 06:12