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Topic: another boring critisism of Cantor's Theorem
Replies: 88   Last Post: May 3, 2004 6:56 AM

 Messages: [ Previous | Next ]
 Rupert Posts: 3,810 Registered: 12/6/04
Re: another boring critisism of Cantor's Theorem
Posted: Apr 16, 2004 10:22 PM

"Herman Jurjus" <h.jurjus@hetnet.nl> wrote in message news:<c5o47o\$3ei9o\$1@ID-110648.news.uni-berlin.de>...
> "Rupert" <rupertmccallum@yahoo.com> wrote in message news://d6af759.0404141902.4ca869e@posting.google.com...
> > david_lawrence_petry@yahoo.com (David Petry) wrote in message news:<25bac3c0.0404141222.7c584b80@posting.google.com>...
> > > rupertmccallum@yahoo.com (Rupert) wrote
> > >

> > > > david_lawrence_petry@yahoo.com (David Petry) wrote
>
> > > > > Could you explain why V can't be countable? Certainly you can't
> > > > > prove that in any consistent first order formalism for which the
> > > > > Loewenheim Skolem theorem applies.

> > > >
> > > > It's a triviality to prove V can't be countable in ZFC. The
> > > > Loewenheim-Skolem theorem says that if ZFC is consistent, it has a
> > > > countable model. But that's not V.
> > > >
> > > > Philosophically speaking, if we are discorsing in the first-order
> > > > language of set theory and uttering theorems of ZFC, we can always
> > > > suppose, without making any of our theorems false, that our discourse
> > > > is relative to a countable model. But as I say, that's different to
> > > > saying V can be countable.

> > >
> > > Somehow, I don't think you explained anything. You merely restated
> > > what you said the first time.
> > >
> > > If you can prove V exists, you can prove that it is countable
> > > relative to some meta-language. But you claim otherwise. Why?
> > >
> > > I suppose I should admit that I don't know a great deal about
> > > this topic, but you have hardly increased my knowledge of it.

> >
> > If ZFC is consistent, and we are uttering theorems of ZFC in the
> > first-order language of set theory, we can always entertain the
> > possibility, without making any of our theorems false, that our
> > discourse, let's call it the object language, is relative to some
> > countable model. Then a metalanguage will be available in which what
> > we referred to as V in our object language, will now in fact be
> > countable. This possibility *may* hold of our language, I don't think
> > it *has* to hold, this may be a point of difference between us.
> >
> > But in any case, you have to decide what language you're talking in.
> > Whether you're talking in the object language or the metalanguage, it
> > won't be true that V is countable. In both the object language and the
> > metalanguage, all theorems of ZFC are true, and it's a trivial theorem
> > of ZFC that V is not countable.

>
> But then, just like the original thing only _seemed_ uncountable,
> but in reality countable, your 'meta world' could also be only
> seemingly uncountable, but be 'meta-meta' countable.
> Or do i miss something?
>
> Cheers,
> Herman Jurjus

We can always entertain the *possibility* of any language that its
semantics is relative to a model which is correctly called "countable"
in some metalanguage. I don't think this possibility always has to
hold.

Date Subject Author
3/29/04 Craig Feinstein
3/29/04 magidin@math.berkeley.edu
3/29/04 A N Niel
3/29/04 Craig Feinstein
3/29/04 Will Twentyman
3/30/04 Robin Chapman
3/30/04 Jose Carlos Santos
3/30/04 David McAnally
4/3/04 Richard Sabey
3/31/04 Jesse F. Hughes
3/31/04 Lee Rudolph
3/31/04 Jesse F. Hughes
3/31/04 Eckard Blumschein
3/31/04 Jesse F. Hughes
3/31/04 Lee Rudolph
3/31/04 Jesse F. Hughes
3/31/04 Detlef Mueller
4/1/04 Eckard Blumschein
4/1/04 Detlef Mueller
4/1/04 Eckard Blumschein
4/1/04 Herman Jurjus
4/2/04 Eckard Blumschein
4/2/04 Hermann Kremer
4/5/04 Eckard Blumschein
4/6/04 Hermann Kremer
4/7/04 Eckard Blumschein
4/2/04 Hermann Kremer
4/6/04 Eckard Blumschein
4/1/04 ZZBunker
4/1/04 David McAnally
4/1/04 Eckard Blumschein
4/1/04 Robin Chapman
4/1/04 Hermann Kremer
4/2/04 Eckard Blumschein
3/31/04 Robin Chapman
3/31/04 Hermann Kremer
4/1/04 Jesse F. Hughes
4/1/04 Hermann Kremer
4/1/04 Eckard Blumschein
3/31/04 Thomas Nordhaus
4/1/04 Detlef Mueller
4/1/04 Eckard Blumschein
4/1/04 Hermann Kremer
4/2/04 Eckard Blumschein
4/2/04 Jesse F. Hughes
4/1/04 Justin Davis
4/2/04 Eckard Blumschein
4/2/04 Robin Chapman
3/31/04 David McAnally
3/31/04 David Petry
3/31/04 Torkel Franzen
4/2/04 David Petry
4/2/04 Torkel Franzen
4/1/04 Rupert
4/2/04 David Petry
4/2/04 David C. Ullrich
4/3/04 Torkel Franzen
4/3/04 David C. Ullrich
4/4/04 David Petry
4/4/04 Daryl McCullough
4/5/04 Rupert
4/5/04 David Petry
4/6/04 Rupert
4/6/04 Rupert
4/7/04 David Petry
4/7/04 Rupert
4/9/04 Herman Jurjus
4/9/04 Rupert
4/13/04 David Petry
4/13/04 Rupert
4/14/04 David Petry
4/14/04 Rupert
4/14/04 Rupert
4/16/04 Herman Jurjus
4/16/04 Rupert
4/22/04 Eckard Blumschein
5/3/04 Herman Jurjus
4/6/04 Herman Rubin
4/6/04 David Petry
4/7/04 Rupert
4/7/04 Rupert
4/5/04 David C. Ullrich
4/6/04 Herman Rubin
4/3/04 Rupert
3/31/04 Ted Hwa
3/29/04 David Manheim
3/29/04 Jesse F. Hughes
3/29/04 Lee Rudolph
3/29/04 Jesse F. Hughes