Re: (Not quite) Cantor's diagonal proof

From: George Greene (greeneg_at_cs.unc.edu)
Date: 10/26/04 

Date: 26 Oct 2004 11:54:20 -0700

Norman Megill <nm@nospam.see.signature.domain.invalid> wrote in message news:<Fe6dnXGLUO_hc-rcRVn-gg@rcn.net>...
> George Greene wrote:
>
> > Robin Chapman wrote
> >
> >>Surely a more important result than the uncountability of R
> >>is that R has cardinality 2^aleph_0.
> >
> > That is a *definition*, NOT a "result".
>
> 2^aleph_0 is not defined as the cardinality of R.

Come on. I'm on YOUR side.
The relevant point here is not so much that 2^aleph_0 "is not
defined as the cardinality of R" as it is that 2^aleph_0 IS NOT
the cardinality of R. My point was that whatever it is, it is
in virtue of DEFINITIONS and not of more complicated considerations.
Cardinality in the ZFC realm is defined in terms of bijections.
BY DEFINITION, two sets have the same cardinality if a bijection
exists between them. You can also by definition get cardinalities
being < or > on the basis of the existence of injections or surjections.
ALL OF THESE ARE DEFINITIONS AND NOT theorems. That was the point.

> There is a general definition for cardinal exponentiation,

Which is irrelevant.
The relevant operator in the sentence we were arguing about was
^,
which is NOT limited in application to cardinals.
It takes sets generally.

> into which you plug "2" and "aleph_0".

BOth of those are ORDINALS first and cardinals SECOND,
derivatively, by convention.

> That this specific instance of cardinal exponentiation
> evaluates to the cardinality as R is a somewhat difficult proof, and
> I haven't worked it out formally yet.

Neither has anyone else, nor will they ever, since it's already
been proven that ZFC does NOT impute a unique cardinality to 2^aleph_0.
There are different models of ZFC with different cardinalities for
2^aleph_0. Cardinalities and the associated alephs ascend via
LIMIT ORDINALS, NOT via powersets!

>
> > For all practical purposes,
> > 2^X *means* "The powerset of X".
>
> Well, it is not a completely trivial proof. That 2^X is equinumerous to
> the powerset of X is proved here:
> http://us.metamath.org/mpegif/pw2en.html (warning: 400K page)
> But yes, once it is proved, for practical purposes they mean the same thing.
>
> > More to the point, 2^aleph_0 actually IS NOT
> > a cardinality. I mean, it HAS a cardinality,
> > but nobody knows what that cardinality is.
>
> 2^aleph_0 is a cardinal number,

No, it isn't.

> since cardinal exponentiation of two
> cardinal numbers is a cardinal number (just like the addition of two
> integers is an integer, etc.). What you mean is: you can't know
> whether it equals aleph_1 in the absence of CH.

Or whether it equals anything else either.
More to the point, it is PROVABLE that 2^aleph_0 is NOT
any PARTICULAR cardinal number, since you can model it
with any of many DIFFERENT cardinal numbers.
More to the point, though, CARDINAL NUMBERS DON'T EXIST.
When we want to pretend that they do, we just conventionally
take initial ordinals as cardinals. Given that everybody
knows that this is what is happening, allegations like
"2^aleph_0 is a cardinal number" are just fatuous.
Ordinal arithmetic is what's relevant. Cardinals are
a VERY thin branch of what's actually going on and NOT
an important one.

>
> > They just know that it has to be bigger than
> > aleph_0.
>
> Yes. And also that it is at least as big as aleph_1.
> http://us.metamath.org/mpegif/aleph1.html
> and thus strictly bigger than aleph_0 by
> http://us.metamath.org/mpegif/alephordi.html
>
> > You can't prove a definition.
> >
> > What he *needed* to be proving was the general result
> > that there cannot be a bijection between a set and
> > its powerset.
>
> Already done.

Of course.
But my point is simply that it is not
what he was focusing on.

> Cantor's theorem:
> http://us.metamath.org/mpegif/canth2.html
> which implies no equinumerosity:
> http://us.metamath.org/mpegif/sdomnen.html
> which implies no bijection:
> http://us.metamath.org/mpegif/bren.html

Please note the PRIOR occurrence of the word
"Cantor's" in the title of the thread. OF COURSE
it's all already been proven. The point was that it
had not yet been properly FOCUSED ON by the person
initiating this inquiry!

Relevant Pages

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