On Jun 16, 3:34 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > On 16 Jun., 21:40, MoeBlee <jazzm...@hotmail.com> wrote: > > > On Jun 16, 2:21 pm, MoeBlee <jazzm...@hotmail.com> wrote: > > > > I don't have a thousand lifetimes to wait for you to show a > > > formula P such that both P and ~P are derivable in ZFC from the above > > > definition. > > > CORRECTION: I should have said: > > > [...] to show, for some formula P, a proof in ZFC (with said > > definition included) of P and a proof in ZFC (with said definition > > included)of ~P. > > 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".
> 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.
> 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").
If in some other sense you find ZFC not compatible with your own mathematical or other notions, is another matter.
A technical note: It is conceivable that one could prove in ZFC that ZFC is inconsistent but as one does not show a particular derivation of a formula "P & ~P" (I think),i.e., showing "non-constructively" that ZFC is inconsistent. But still if ZFC is inconsistent then there does exist a constructive proof that ZFC is inconsistent, which could be picked out from an enumeration of the proofs (though, of course, it is recognized that this may not be feasible within the presumable lifetime of any human being).
A note on the discourse: If you persist to present me with misunderstandings of ZFC and arguments and claims using terminology not defined to primitives and based in axioms in some stated formal language and logic (even if only informal arguments that nevertheless suggest some background formal langauge, logic, and axioms) then I may leave many, perhaps all, such arguments without my response, since I do need to start better valuing my time than to fruitlessly try to bring you to reason and understanding. This is not to say that I reject philosophical discussion or even yet-to-be formalized mathematical discussion; but rather that I don't wish to be mired in conversation where such as just mentioned are conflated with mathematical argumentation that at least in principle admits of resolution per some formal theory or another.