(November 26, 2022 at 7:41 pm)Jehanne Wrote:(November 26, 2022 at 7:19 pm)polymath257 Wrote: ZFC vs naive ST isn't so much the issue here as clarity about what an infinite decimal expansion means. Which means being a bit more precise about what it means to be a real number (although, in this case, we are doing convergence in the rationals).
Real numbers, of course, include irrational numbers. In my discrete mathematics course (Rosen was the text), ZFC was mentioned but not taught. Professor Rosen has a text on elementary number theory; I never took that class, as I was a computer science major. Some of the concepts in number theory were mentioned in my automata theory class.
About the only thing that needs to be said is that naive set theory is quite sufficient until it isn't. Very few people ever need to deal with the axioms of Foundation or Replacement and their consequences for the iterative hierarchy. The vast majority of mathematics happens below level omega+20 of the hierarchy.
The trickiest axiom for most people is the Axiom of Choice, which has a long history of controversy. It is incredibly easy to use AC without realizing it and even PhD mathematicians are often uncertain about it.
That said, it is usually just assumed without comment when studying Turing machines and Godel's Theorems.
