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


Re: Mathematics as a language
Posted:
Nov 5, 2010 1:02 AM


Marshall wrote: > herbzet 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?
Seems that way to me  which also says to me that the axioms of these theories aren't sufficiently strong to compel a choice between the models of an undecidable sentence and the models of its negation  all of which exist.
(All this presumes the theory is consistent  an inconsistent (classical) theory contains no undecidable statements, as it will blithely prove every statement in the language.) (As a side issue, whether "most" theories have undecidable sentences or not, I guess there's an infinite number of both kinds.
The broadest characterization I know of for theories which are not only incomplete but essentially incomplete (that can't be made complete by adding axioms) is that any consistent extension of Robinson arithmetic (a fragment of PA) is essentially incomplete. But I'm guessing that this does not exhaust the category of essentially incomplete theories.)  hz

