[FOM] Infinity and the "Noble Lie"
joeshipman@aol.com
joeshipman at aol.com
Thu Jan 5 02:37:47 EST 2006
Haney:
It seems to me that the question of the "truth" of the axiom of
infinity is almost entirely ridiculous, considering that the axiom is
such an extremely idealized version of real-world experience from which
the axiom is derived. And since the motivation for the question seems
ultimately to determine what is the value of the axiom, it seems to me
that the appropriate question is whether the axiom is *useful*.
I reply:
Are you prepared to say that the question of the "truth" of an
arithmetical statement proved using the axiom of infinity is also
ridiculous?
Consider Ramsey's theorem -- For all a, b, and c, there is an N such
that if the a-element subsets of {1,2,...,N} are b-colored, then there
is a monochromatic subset of {1,2,...,N} of size >= c (that is, all of
its a-element subsets received the same color).
This can be proven by a compound induction, with an explicit primitive
recursive bound for the function N(a,b,c), within "finite mathematics".
Now consider the Paris-Harrington Theorem, which changes the conclusion
of Ramsey's theorem to require that the monochromatic subset S be
"relatively large" (|S|>min(S)). All proofs of this theorem must assume
the axiom of infinity.
You have only 3 options:
1) refuse to assent to the "truth" of Ramsey's Theorem
2) say that Ramsey's Theorem is "true" but deny that such a status can
be granted to the Paris-Harrington Theorem
3) say that the Paris-Harrington Theorem is "true" and explain how a
"true" statement can depend in an essential way on an axiom whose
"truth" is a ridiculous question. If you are going to say that the
concept of "truth" is not relevant to the Axiom of Infinity, you need
to say what "true" axioms you think can be used to establish the
Paris-Harrington theorem as "true".
Which will it be?
-- JS
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