Re:How many real numbers exist? New proof moves closer to an...
by graycat on Friday July 16, 2021 @04:01AM
(#27853947)
The Banach-Tarski paradox is fine with me: Not all subsets of the real line or R^n for the set of real numbers R and positive integer n are measurable. Yes, the usual example of a non-measurable set uses the axiom of choice. The sets in the Banach-Tarski paradox are not measurable -- okay.
Of course the natural numbers can be put into 1-1 correspondence with the rationals -- how to show that is the classic Cantor diagonal argument.
At a tea before a seminar, I asked Max Zorn what Paul Cohen had just proved. Zorn didn't explain it and instead just loaned me his copy of Cohen's paper -- I lost it in a move! If Zorn wasn't strongly interested in Cohen's proof, then neither should I be.
For any set of axioms, there will be statements we can neither prove nor disprove. Surprising, interesting, got it.
Zermelo-Fraenkel set theory with the axiom of choice seem fine to me -- now back to the real work!
Of course the natural numbers can be put into 1-1 correspondence with the rationals -- how to show that is the classic Cantor diagonal argument.
At a tea before a seminar, I asked Max Zorn what Paul Cohen had just proved. Zorn didn't explain it and instead just loaned me his copy of Cohen's paper -- I lost it in a move! If Zorn wasn't strongly interested in Cohen's proof, then neither should I be.
For any set of axioms, there will be statements we can neither prove nor disprove. Surprising, interesting, got it.
Zermelo-Fraenkel set theory with the axiom of choice seem fine to me -- now back to the real work!