On 16 Jun., 23:43, MoeBlee <jazzm...@hotmail.com> wrote:
> > Look simply at the results. > > If a theory says that there is an uncountable set of real numbers such > > each number can be identified as a computable or definable or > > constructable one, or in other ways, then this theory is provably > > wrong. > > Please tell me which EXACT, SPECIFIC theorem of ZFC "says" that "each > real number can be identified as a computable or definable or > constructable one, or in other ways".
If you start with Cantor's list, then the diagonal can be identified. Do you mean that in ZFC Cantor's argument is wrong?
Further: A number that cannot be put in trichotomy with others is not a number. Is such an unknown entity called a number in ZFC? > > > Reason: Cantor either proves that a countable set is uncountable or > > that a constructible/computable/definable number is not constructible/ > > computable/definable. > > Whatever Cantor proves, my question is specifically as to ZFC.
I consider results only. If ZFC allows the conclusion that Cantor's diagonal argument is wrong, then ZFC may be right. Is it right? > > > What the theory internally may be able to prove or not to prove is, at > > least for my person, completely uninteresting. > > What interests you is entirely up to you, of course. But you've not > shown that ZFC is inconsistent (where 'inconsistent' is defined as > "having as a theorem some formula and its negation").
I have been told that ZFC yields this result: There are uncountable many real numbers. And by number I understand something that can be put in trichotomy with other numbers. If this is not the case, ZFC may be right. Others have told me that ZFC does not prove any uncountability at all and that the "standard model" is not a model of ZFC at all. If this is the case, ZFC may be right.