I recently have been studying the axiomatic construction of the set of real numbers through the Peano axioms for the natural numbers. It seems to me, the only things needed to proceed with this body of knowledge, is some basic set theory (ZFC axioms) plus the rules of mathematical logic. However, I noticed that from time to time one needs the facts mentioned in the title. Now, these are self-explanatory and self-evident of course but since I am so deep in the roots of modern mathematics, I thought I should inquire about those as well. After all, Euclid states a similar axiom in his Elements: «if each one of two line segments is equal to a third one, then they are equal.»
So, regarding real numbers, where did these rules come from?
- Are they axioms of some sort and if yes from which theory?
- Are they simply some statement formulas of boolean logic? If so, how is one to be convinced of their validity?
Thanks so much.