[FOM] infinity and the noble lie
joeshipman at aol.com
Mon Jan 9 14:05:37 EST 2006
I'm finding this whole discussion confusing, and I think that is not
entirely my fault. In particular, I think there are many formal and
nonformal issues which are much more subtle than this discussion is
letting on. Consider the following from Joe Shipman:
"A theorem cannot be MORE certain than the axioms it is derived
from. Therefore, if you won't call the set-theoretical axiom of
Infinity "true", you had better explain whether you are willing to
call the Paris-Harrington Theorem "true". If you are so willing, you
should be able to identify "true" axioms it can be proved from."
Well of course a theorem can be more certain than the axioms it is
derived from. Let P&Q be an axiom and P a theorem. What is
generally true is that a theorem can't be LESS certain than the
axioms it is derived from. So what is going on here?
Although it is true that a STATEMENT may be more certain than the
axioms it is derived from, a theorem, AS A THEOREM, may not be. If you
say the Paris-Harrington Theorem is somehow more "true" or more
"certain" than ZFC's Axiom of Infinity, the derivability in ZFC of PH
from AxInf cannot be the explanation for the confidence with which PH
is asserted; if you give PH ANY greater degree of certainty than
AxInf, you must explain why -- for example, by proving PH from a more
acceptable axiom and explaining why it is more acceptable.
Another possibility would be to say something like "PH is more certain
because of its FORM as a Pi^0_2 arithmetical sentence, and I regard all
Pi^0_2 consequences of the Axiom of Infinity as 'true' statements".
But this is not adequate as an *explanation* of the epistemological
disparity between AxInf and PH.
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