Marshall
Posts:
1,936
Registered:
8/9/06


Re: Mathematics as a language
Posted:
Nov 4, 2010 11:10 AM


On Nov 3, 11:40 pm, herbzet <herb...@gmail.com> wrote: > 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.
Ok. But again, all the "counterexamples" just amount to saying that (most) theories have undecidable sentences, right?
> (Excuse me for belaboring the point, but Marshall is, after all, > a programmer.)
Yay!
Marshall

