On Jun 17, 1:54 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > 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.
You have a very irritating habit of jumping the conversation to deny some claim I never made. This is about the third time you've done it in this thread.
I never said a real number can be defined by an infinite sequence. (Though I don't preclude treatments in which such things are done.)
> A real number > can be defined only by a finite word. But there is no diagonalization > over finite words.
Without even commenting on what you mean or whether it is true, it does not refute that the formalized argument is first order logic applied to axioms and incontrovertible that it (the formalized argument, i.e., the proof) exists (exists AS a PROOF from certain axioms with a certain logic).
> > 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.
You've not said what "wrong" assumption I've "started with".
As to your contradiction, again I'm not very interested in your own renderings of mathematical results nor, in this regard, Cantor.
All of this business of yours does not refute what is simply introvertible, that a formal proof exists in the manner I've mentioned.
> The translation of these notions into your "incontrovertible" theory > is the weak point.
NO, you did not listen to what I said. I did NOT say anything about an incontrovertible THEORY. Rather, I said it is incontrovertible that a certain finite sequence of finite sequences of symbols exists.
Please address what I have said and not posture is if you are commenting on something I said when in fact I didn't say it.
> > > 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?
I don't know. What is the definition IN THE LANGUAGE of ZFC of 'constructible number'?
Anyway, I have no idea how you think that bears upon what I just wrote.
> Is Cantor's diagonal not a constructible number?
This reveals your misunderstanding of the very basics of Cantor's argument (which I prefer to take in it's formalization in Z set theory).
There is no object in ZFC that is "Cantor's diagonal".
Rather, for any given enumeration of a set of binary sequences, there is a diagonal and anti-diagonal for that enumeration.
So, your question should be, for any given enumeration of a set of denumerable binary sequences, is the diagonal and/or the anti-diagonal constructible?
Or you might mention some particular enumeration of some particular set of denumerable binary sequences and ask whether its diagonal and/ or anti-diagonal is constructible?
But the answer would depend on a *ZFC* definition of 'constructible'.
> > > > > 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?
I usually do. ASCII though sometimes makes it tough.
In any case, you've taken my point about your stand-alone predicate 'number'.
> Cantor > uses some x of R and shows that R is an uncountable set by > constructing an x_0 of R.
The Cantor method I am familiar with is this (it does not rely upon real numbers actually being denumerable binary sequences):
Given a denumerable enumeration f of a set of denumerable binary sequences, there is the "diagonal" ('diagonal' not to be taken LITERALLY as a geometric object necessarily) of that enumeration and that diagonal is itself a denumerable binary sequence. Then there is the anti-diagonal from the diagonal, and the ant-diagonal is a denumerable binary sequence not in the range of f. Thus for any enumeration f of a set of denumerable binary sequences, f is not onto the set of all denumerable binary sequences. Thus the set of all denumerable binary sequences is not countable.
Also we show that there is a bijection between the real interval [0 1] and the set of denumerable binary sequences. So [0 1] is uncountable.
And if R is countable then, a fortiorti, [0 1] is countable.
Are you or are you not familiar with various certain ordinary Z theory formalizations of Cantor's argument?
Please answer that question. I cannot address you very well on this subject if I don't know whether you are or are not familiar with some treatment in which Cantor's argument is formalized in Z set theory.
> We can show, probably also in ZFC, that the constructible elements of > R are not an uncountable set.
That requires your definition IN THE LANGUAGE of ZFC of 'constructible'.
Do you know what is meant by 'the language of ZFC'?
Please answer that question, as again, I need to know in order to guage my discussion with you.
> > > 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?
Actually, there is another proof in which the anti-diagonal method is not used.
And again, just to be clear, there is not THE anti-diagonal, but rather there is the anti-diagonal for each given enumeration of a set of denumerable binary sequences.
> So, concerning the constructible elements alone, ZFC cannot able to > prove their countability? Or how else could you avoid a contradiction?
Please, you're wearing my patience.
I have no doubt that you may show whatever contradictions between ZFC and your own notions. That is not at issue with me.
On the other hand, you have not shown any ZFC derivation of a formula P & ~P nor have you shown that one exists.