[FOM] Infinity and the "Noble Lie"
Patrick Caldon
patc at cse.unsw.edu.au
Mon Dec 12 01:04:09 EST 2005
> I think this whole discussion is doomed. Let me say some final words:
> The only thing that needs to be finite is your argumentation to convince
> me of a theorem. For the rest there is no limit.
Tangentially to the discussion:
You're not happy to accept Gold "identification in the limit" style
arguements? Suppose I've a (possibly infinite) set of natural numbers
in mind, and you want to know the gcd of those numbers. I tell you 20
is in the set - you guess 20 as the gcd. I tell you 10 is in the set
- you change your guess to 10. I tell you 40 is in the set - you keep
your guess at 10. It's clear that your guess will converge to the
gcd of my set - but provided I never tell you two co-prime numbers,
you'll never be able to give me the number "1" with certainty.
Our argumentation is not finite - but there will there will be some
statement of the form "k is the gcd of the numbers you have in mind"
that you will produce co-finitely many times, and indeed k will be the
gcd of this set. Is "k is the gcd of the numbers you have in mind" a
theorem?
Patrick.
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