Learning the foundations of math

jsalvatier

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.

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.

So it looks like you're interested in learning the fundamentals of classical logic and set theory, and then paving your way towards measure theory. If you don't have a solid background in set theory, then you probably don't have a strong background in real analysis either, which to my understanding is needed for measure theory. (I don't know how you can do measure theory without even knowing what the Riemann integral is).

You should take a look at Real mathematical analysis by Pugh. You're going to need to know basic stuff like basic set theory and functions and such as a prereq, but it's a very lucid introduction to real analysis with metric space topology and compactness and connectedness and all that. It was the first book I found that had a good explanation of Dedekind cuts (Rudin's Principles of Mathematical Analysis has a very terse description. Pugh, on the other hand, has pictures!)

For measure theory, I haven't read very far through them but here is a 4-volume set on Measure theory available for free online from D.H. Fremlin. It has some introductory set theory stuff if I recall, which might be a good starting point if your grasp on logic is strong enough.

For basic logic, How to Prove It: A Structured Approach by Velleman is my personal favorite.

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Learning the foundations of math

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Learning the foundations of math

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