
Re: Mathematics as a language
Posted:
Nov 4, 2010 7:58 AM


Bill Taylor says... > >On Nov 4, 3:12 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > >> >What logically could exist  that is, what is not inherently self >> >contradictory  has mathematical existence. >> >> The problem with this is that there could be two different >> mathematical objects, A and B, such that neither is inherently >> selfcontradictory, but the existence of A contradicts the >> existence of B. They can't, therefore, both exist. > >Exactly so! > >My favorite example being, >(as someone just now pointed out half of), >it is consistent with ZF that > >1) there exists a set of cardinality strictly between N & R; > >2) there exists a function on P(R) whose values > are bijections between the argument and elements of N u {N,R}.
Thanks! That's a much better example than mine, because it's much more concrete.
>1 & 2 are individually consistent with but jointly inconsistent >with ZFC. (1) means CH is false, (2) means CH is true. > >I once asked about this of Thomas Forster, an admirable logician, >who often claims informally that "consistency is existence". > >He refused to answer, saying only that "ontology is poisonous". > >I took that as meaning I'd won the argument! :)
No, it means that you're a snake. A poisonous one.
And I mean that in a good way.
 Daryl McCullough Ithaca, NY

