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.
Principles of Arithmetic and Geometry by Carl Allendoerfer
If you are looking for a book that begins before even numbers begin, then this is it.
The book starts with basic logic, then moves to define basic set theory very clearly, then proceeds to explain how set theory is the foundation of our whole number system, and finally moves to defining all elementary operations using what you've learned. He goes into great depth of defining all the various kinds of correspondences, cardinality, and other crucial ideas that usually get ignored early on.
Allendoerfer stresses the importance of developing abstract thinking and really makes you feel confident and excited about the abstract nature of mathematics, rather than hinging to real world examples. The truth is that, in order to be a useful mathematician you must be comfortable and skilled with abstract thinking. Whereas most professors shy away from this and want you to wait until much later to develop these skills, Allendoerfer makes it very clear that, no matter what your age is or experience, you are ready to think abstractly about nature and about mathematics.
Every chapter is full of exercises with all answers provided, as well as a test at the end of each chapter and recommendations on what to do if you struggled.
I have yet to find any book that does such a good job at explaining these foundations as Allendoerfer. This book is a god-send for those of you out there that are trying to self-teach mathematics, especially modern mathematics, and are willing to begin at the beginning. (Arithmetic is NOT even the beginning!)
When this book is completed, I highly recommend his Freshman Mathematics book and also his Principles of Modern Mathematics.
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):