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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