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?
2026-04-06 19:03:06.1775502186
Are there any recent (past 5 years or possibly upcoming) texts or sources focused on proofs/theory regarding undecidable theorems?
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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.