Foundations of Mathematics - Where to Start?

Question

I am interested in the foundations of mathematics. However, I don't know where to start.

From what I've read, the most popular foundational system for mathematics is Set Theory: ZFC specifically. ZFC is formulated using First-Order Logic. So, from these two things the obvious starting point for me is:

1. Learn (at least the basics of) First-Order Logic.
2. Start learning about ZFC.

My question: What are the best (relatively advanced) textbooks for 1 and 2? A textbook combining 1 and 2 would be ideal.

Answer 1

A good starting point is Halmos's Naive Set Theory. It is "naive" in the sense that it does not use formal logic, so it may not satisfy your (1). However it is a standard textbook in the field and very readable (and short), so I will recommend it. Note that to formalize ZFC you will actually need second-order logic.

Answer 2

For first-order logic, one standard reference is Enderton. I'm also a fan of Boolos, Burgess, and Jeffrey, specifically chapters $9$ and $10$ (this is the text I learned from); incidentally, this book has a lot of material on computability theory which you can ignore for your purposes, although personally I recommend reading it as well (admittedly, perhaps skipping chapters $5$ and $6$).

For set theory itself, my favorite text is Kunen, but it's pretty dry. You may prefer something like Ciesielski.