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Re: [ontolog-forum] Axiomatic ontology

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Chris Menzel <cmenzel@xxxxxxxx>
Date: Thu, 7 Feb 2008 10:18:11 -0600 (CST)
Message-id: <alpine.DEB.1.00.0802071006070.14396@xxxxxxxxxxxxxxxxx>
On Wed, 6 Feb 2008, John F. Sowa wrote:
> Avril,
> ...
> On the other hand, I would not recommend the following approach
> for number theory:    (01)

To say the least.    (02)

> > Axiom 9. can be maintained, but the meaning of 'every' has to be
> > interpreted to denote a totality of something around 10**120,
> > or Ackermann(5 5), or something finite that can be written down
> > or understood in some way.    (03)

The view isn't even coherent.  If Ackerman(5 5) exists, why not
Ackerman(Ackerman(5 5) 5) -- a massively larger number?  And of course
if *that* number exists, well, you get the idea.    (04)

> Many 19th century mathematicians strongly objected to that way of
> talking, and I sympathize with them.  But those mathematicians would
> *never* agree to a fixed upper bound on the integers, such as 10**120,
> Ackermann(5 5), or any other finite integer.    (05)

Indeed -- which of course means that there are infinitely many finite
integers, and hence that there is a set that contains them, hence a
power set of that set, and off we go down the Cantorian bunny trail! :-)
You may not like where that leads, but it is very hard to argue that
there is a nonarbitrary point at which you can stop that line of
reasoning.    (06)

-chris    (07)

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