Re: R^R = R^2 = R

Dave Seaman a écrit :
On Thu, 29 Nov 2007 00:44:24 +0100, Michal Przybylek wrote:
"Dave Seaman" <dseaman@xxxxxxxxxxxx> wrote:

thus if R = aleph_1 then R ^ R > = aleph_2
The "if" clause is not satisfied in ZF, which is the set theory that I
and most others are using.

Huh?

Clearly, the implication is false. Your response too.


The "if" clause is false; it is not the case that R = aleph_1, at least in
ZFC without CH.

Not exactly: the if clause *may* be false



Since the "if" clause is false, it follows that the implication is
vacuously true.

No, it is so only in ZFC + non -CH. But the implication isq still always true, because of Cantor's theorem.



There was nothing wrong with my response.


.

Relevant Pages

  • Re: Riemanns zeroes
    ... Any statement of the form "if RH then X" will be vacuously true ... if RH fails. ... Dave Seaman ... Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling. ...
    (sci.math)
  • Re: R^R = R^2 = R
    ... ZFC without CH. ... Since the "if" clause is false, it follows that the implication is ... vacuously true. ...
    (sci.math)
  • Re: Mathematics made simple.
    ... this one's actually a theorem of ZFC! ... we note that the antecedent says: ... And since there is no universal set ... this one is vacuously true! ...
    (sci.math)