Here lives my introductory text about naive set theory and the mathematical way of writing arguments. It has minimal prerequisites and should work for self study, but the experience will be better if you are able to get feedback when you do the exercises. This could also be suitable for an introductory proofs course in an undergraduate math program, which is how I used it as an instructor.
This text is intended to be a first treatment of foundations in math. It uses logic and set theory to introduce mathematics and mathematical writing, but it is not about metamathematically studying foundations. If you know enough to care about the difference between an axiom and an axiom schema, then this might not be what you're looking for. If you are eager to transition from "lower division" undergraduate math to "upper division" proof-based math, then this text might be for you.
book , sourceExcerpt:
We need to find a good balance between rigor and efficient communication. Finding that balance is one of the great challenges when you first learn how to write proofs. If we sacrifice too much rigor in an argument, then we can lose confidence in its correctness. Or worse, we can start to prove false things! When developing a new mathematical theory, it’s generally good to err on the side of being more rigorous. Then, as the theory develops and common patterns of arguments become routine, one can slowly relax the rigor in favor of efficiency.
Proofs and Mathematical Writing by Ebrahim Ebrahim is licensed under CC BY 4.0