Re: Cantors proof

From: Jesse F. Hughes (jesse_at_phiwumbda.org)
Date: 09/27/04 

Date: Mon, 27 Sep 2004 20:59:31 +0200

Dave Seaman <dseaman@no.such.host> writes:

> On 27 Sep 2004 10:17:23 -0700, Daryl McCullough wrote:
>> Dave Seaman says...
>
>>>Proving that natural numbers are finite is an
>>>exercise in circular reasoning.
>
>> Besides the Dedekind definition, here's another definition
>> of finite that doesn't involve the naturals:
>
> Yes, it does.
>
>> A set X is finite if there is a well-ordering of X whose
>> inverse is also a well-ordering.
>
>> where a relation R(x,y) is defined to be a well-ordering if
>> it is a total ordering such that every set has a R-least element.
>
> Definition. A natural number is a transitive set that is well ordered by
> the set-membership relation and also by the inverse of the set-membership
> relation.

I'm confused about what ya'll are taking as given here.

I thought that the argument was whether it is tautological that
elements of any model of PA are "finite". If that's what you're
talking about, it seems odd to define natural numbers as particular
sets.

I guess that elements of nonstandard models of PA do not satisfy your
definition, right? (What would it mean anyway?) 

-- 
"But he himself was not to blame for his vices. They grew out of a personal
defect in his mother. She did her best in the way of flogging him while an
infant... but, poor woman! she had the misfortune to be left-handed, and a
child flogged left-handedly had better be left unflogged." -- E.A. Poe
Quantcast