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.

The meta-theory parts, so that I am learning just how to make proofs in theory X (e.g. propositional logic), and not learning how to prove things things about theory X proofs. Introduction to Mathematical Logic claims that all theories can be formalized; learning how to work in a theory first and then later possibly coming back to learn how to prove things about proofs in that theory seems like a good way to avoid being confused, and that's largely my goal. Does that clarify?

That depends on what you want to use formal logic for. If you just want some operational knowledge of propositional logic for working with digital circuits, then yes, any digital systems textbook will teach you that much without any complex math. Similarly, you can learn the informal basics of predicate logic by just figuring out how its formulas map onto English sentences, which will enable you to follow its usual semi-formal usage in regular math prose. But if you want to actually study math foundations, then you need full rigor from the start.

Perhaps t... (read more)

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Learning the foundations of math

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Learning the foundations of math

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