In article <email@example.com>, JRStern <JRStern@foobar.invalid> writes: >On Sun, 28 Oct 2012 07:31:24 +0000 (UTC), "Peter Webb" <webbfamilyDIEspamDie@optusnet.com.au> wrote:
>>> I suppose at worst it could be made an optional axiom, or some >>> statement of principle could be added as an additional axiom to allow >>> it to be true. >>> >>Its a theorem in ZF. It doesn't need an axiom. > >It seems a little involved to be the kind of thing we like to call a >theorem,
"It's complicated so make it an axiom, even though it can be proven from the other axioms"? That's certainly an interesting suggestion.
Following that approach, I suppose that it would make sense to add other theorems with complicated proofs to their respective areas as new axioms. For instance, we could add the Four-color Theorem as a new axiom in topology, or Wiles's Theorem as a new axiom in number theory.
However, since these, like the theorem that |P(S)| > |P(S)|, have already been proven, I don't really see the advantage of adding them as axioms. After all, it's not really any more difficult to cite "Four-color Theorem" than it is "Four-color Axiom", is it?
As I understand it, mathematicians usually want to: 1. Have as few axioms as possible. 2. Keep the axioms as simple as possible.
I'm pretty sure that these goals are why various mathematicians in the nineteenth century tried to prove the Parallel Postulate from the other axioms of plane geometry.
-- Michael F. Stemper #include <Standard_Disclaimer> Life's too important to take seriously.