Books that follow axiomatic approach?

Question

What are some maths textbooks that follow the "axiomatic approach"? (I would call it "theorem-proof" approach, but I'm more after books that start from the complete basics in a branch of math)

What I consider "axiomatic approach" books: e.g. Disquisitiones Arithmeticae, Euclid's Elements

Answer 1

Here are some books that include constructions of the real numbers from the natural numbers:

• Number Systems and the Foundations of Analysis by Mendelson
• The Number System by Thurston
• The Structure of Number Systems by Parker

Answer 2

1. Foundations of Analysis by E. Landau.

2. The Foundations of Geometry by D. Hilbert.

3. Galois Theory by D. Cox.

4. Principles of Mathematics by B. Russell.

Answer 3

Anything by Nicolas Bourbaki. Some specific recommendations:

1. Algebra 1 and 2.
2. Commutative algebra.
3. Integration 1 and 2.
4. Lie groups and Lie algebras, 1, 2 and 3.

Bourbaki is very far from being suitable for everyone. That said, if you're looking for a reference that's axiomatic, super-abstract, and works in greatest possible generality, and is also crystal clear and careful, Bourbaki is the place to go.

Also, I'd say that Grothendieck's Éléments de géométrie algébrique is the axiomatic reference for algebraic geometry.