The ability to understand the beauty of math requires rigorous study. However, most people do not have access to the kind of training pure math requires.
Many of my friends easily get interested in the subject when demonstrated beautiful proofs and constructs, but I am at loss when they ask for resources to learn more, since we had never used books in our studies, instead, the teacher would show the art of math by proving theorems and building constructs together with the students using only the whiteboard.
I am looking for books or online courses on that are suitable for beginners (more on this below) on the following subjects:
- Intro to sets, mappings, boolean logic, predicate theory
- Number theory
- Rings, fields, groups, vector spaces, Galois theory, etc
- Set theory, measure theory, functional analysis
- Combinatorics, graph theory
- Language theory, lambda calculus
- Other subjects that are not in this list
Books rigorously building the field entirely from ground up (axioms) with detailed looks at all the important proofs with multiple versions and highlighting relations to other fields would be best. It is good if the book has exercises in the form of main story lemmas and side story / related proofs.
Please share your favorite pure math works below.