On 17 Jun., 19:23, MoeBlee <jazzm...@hotmail.com> wrote: > On Jun 17, 2:49 am, WM <mueck...@rz.fh-augsburg.de> wrote: > > > 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? > > In this particular conversation, my interest is in Cantor's argument > as it is formalized in Z set theory. And of course it's not "wrong". > It is simply first order logic applied to axioms.
No, this application is erroneus. A real number cannot be defined by an infinite sequence. A real number can be defined only by a finite word. But there is no diagonalization over finite words.
> You may wish to have > any feeling or conviction you like about first order logic and the > axioms, but that Cantor's argument can be formalized as proof in Z > set theory is not controvertible. (As I alluded, a proof being a > certain kind of finite sequence of finite sequences of symbols.)
But you start with a wrong assumption. This is proved by the contradiction: Either Cantor proved the uncountability of a countable set or Cantor proves the inconstructibility of a constructible number.
The translation of these notions into your "incontrovertible" theory is the weak point. > > > 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? > > I make no use of a predicate 'is a number' where 'number' is STAND- > ALONE nor does an ordinary formalization of Cantor's argument. > > And I don't know what failure of trichotomy (in ZFC) you might have in > mind.
Then forget it. It is not important. Imporrtant is: Either Cantor proved the uncountability of a countable set (namely the constructible numbers) or Cantor proves the inconstructibility of a constructible number.
> > > > > 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? > > "right", "wrong" are your terminology. I've stated already now my most > basic view of Cantor's argument as formalized in Z set theory. Whether > this or that is, in some other sense, "right" or "wrong" is another > discussion, but not needed to address the mere question of the > theoremhood of a certain formula in the language of ZFC.
Does ZFC not prove that all constructible numbers are countable? Is Cantor's diagonal not a constructible number? > > > > > 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. > > ZFC doesn't make use in this context of a predicate "is a number" > where 'number' is STAND-ALONE. As to 'real number" we may regard that > as single predicate. In light of the formal itself itself, we could as > well write 'realnumber' or 'realschmumber' or 'schmeal schmumber' so > that 'number' has no component meaning onto itself. You should > understand that names of predicates in formal mathematics are not > always compositional, particular with adjectives. I can define "is a > slooper spockle" without committing that 'slooper' and 'spockle' have > independent meaning. This is because English is only an INFORMAL > rendering of certain formal predicates. So, as to the FORMAL theory, > for example, where we say "x is a real number", the actual formula > would be "Rx" where 'R' (or whatever other typographic shape) has been > chosen as the formal predicate symbol ordinarily rendered in English > as 'is a real number'. > > Please tell me whether you understand this. If you find fault in it, > then so be it. I'm not interested in arguing with you about it.
Of course, this is not difficult to understand. You may call these objects as you like. But why not use the common abbreviations? Cantor uses some x of R and shows that R is an uncountable set by constructing an x_0 of R. We can show, probably also in ZFC, that the constructible elements of R are not an uncountable set.
> > > 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. > > ZFC proves a FORMULA that we READ or RENDER in ENGLISH as "there > exists an x such that x is uncountable".
This proof is probably done by constructing another element called the anti-diagonal? So, concerning the constructible elements alone, ZFC cannot able to prove their countability? Or how else could you avoid a contradiction?