herb z
Posts:
1,187
Registered:
8/26/06


Re: Mathematics as a language
Posted:
Nov 4, 2010 2:40 AM


Herman Jurjus wrote: > Marshall wrote: > > Herman Jurjus wrote: > >> herbzet wrote: > >>> Aatu Koskensilta wrote: > >>>> herbzet writes: > >>>>> Bill Taylor wrote: > >> > >>>>>> Or whether the number 6 really exists. Does it? > >> > >>>>> It *could* exist  therefore, mathematically, it *does* exist. > >> > >>>> This is a traditional and appealing idea. But just what is meant by > >>>> "could" here? What sort of possibility is involved? > >> > >>> For rhetorical punch, I purposely left out the modifier, which is "logical". > >> > >>> What logically could exist  that is, what is not inherently self > >>> contradictory  has mathematical existence. > >> > >> Corollary: CH is false. > >> Proof: Since Cohen 1963 we know that it is logically consistent to > >> assume that there exists S, subset of P(N), equipollent neither to N nor > >> to P(N). > > > > Consistent with what? In what theory? > > Coconsistent with ordinary mathematics, of course. > (I.e. with ZFC, and then also with any weaker theory.)
Right  the assumption here is that ordinary mathematics (i.e. ZFC, more or less) is itself consistent  the ordinary and unremarkable gentleman's agreement.
(Excuse me for belaboring the point, but Marshall is, after all, a programmer.)
 hz

