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.

I hope I am not imposing, but Cook's notes have confused me. The first set introduces a syntax which is fine, but then it introduces semantics and starts using several terms that haven't yet been defined (iff, maps and sets) are these part of meta-theory and conceptually different from being part of propositional logic? What am I missing?

Yes, these are concepts from the meta-theory, i.e. the language in which you speak about the formal logic you're defining. When you define, say, sets of formulas, or maps (i.e. functions) from atoms to truth values, these objects exist outside of the formal system (i.e logic) under discussion.

Now of course, you can ask how come we're talking about sets (and functions and other objects which are sets), when we're just defining the formal logic we'll use to axiomatize the set theory. The answer is that you have to start from somewhere; you can't start speak... (read more)

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Learning the foundations of math

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Learning the foundations of math

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