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

  • $\begingroup$ Try the Bourbaki series. $\endgroup$ – Yuval Filmus Oct 23 '14 at 16:23
  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.


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.

  • $\begingroup$ Can you tell me which book(s) of Bourbaki treats abstract algebra? $\endgroup$ – user 170039 Nov 13 '14 at 3:40
  • $\begingroup$ @user 170039: That would be the books titled "algebra." (In serious math, we refer to "abstract algebra" as just "algebra.") $\endgroup$ – Daniel Miller Nov 13 '14 at 3:44
  • $\begingroup$ All the two books, right? $\endgroup$ – user 170039 Nov 13 '14 at 3:51
  • $\begingroup$ @user 170093: yes. $\endgroup$ – Daniel Miller Nov 13 '14 at 3:55

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


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