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. 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.)
> 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.
> > > 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.
> > > 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. But at least you know now how I regard such matters in terminology.
> 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".
As to models, there are different senses of the word 'model' depending on context of discussion.