I have recently become interested in the foundations of math. I am interested in tracing the fundamentals of math in a path such as: propositional logic -> first order logic -> set theory -> measure theory. Does anyone have any resources (books, webpages, pdfs etc.) they would like to recommend?
This seems like it would be a popular activity among LWers, so I thought this would be a good place to ask for advice.
My criteria (feel free to post resources which you think others who stumble across this might be interested in):
- The more basic the starting point the better: I would prefer a resource that defines propositional logic in terms of a context free grammar and an evaluation procedure (don't know if that is possible, but that's the sort of thing I am interested in) to one that just describes propositional logic in English; I would prefer a resource which builds first order logic from propositional logic + some definitions to one that just describes how first order logic works; etc.
- The fewer axioms (perhaps that's not quite the right word) the better. I prefer a resource defines describes propositional logic with just two operators (say negation and conjugation) and then builds the other operators of interest to one that defines it with 5 or 6 operators (I've seen many resources which do this).
- I expect that there are multiple ways to build math from basic building blocks. I am more interested in standard ways than than non-standard ways.
At the risk of performing necromancy, I'd like to point out that the video was highly contested afterward on the IAS Foundations of Mathematics mailing list. Many people thought the talk wasn't a reasonable explanation of the current state of affairs in FOM.
I'm not an expert in this, however.