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). > By your principle above, S mathematically exists. Therefore CH is false.
Our conception of "sets", as codified in ZF(C), if consistent is an incomplete notion that may be extended and sharpened in various directions. All of the resulting structures (we could call them "set universes") have mathematical existence.