[FOM] truth and consistency
parsons2 at fas.harvard.edu
Sun Jun 1 13:16:37 EDT 2003
At 4:01 PM -0700 5/30/03, Martin Davis wrote:
>For higher order systems like type theory or ZFC, I know no reason
>for believing in their consistency other than the fact that the
>axioms are satisfied by our intuitive Cantorian picture of sets of
>sets of sets of .... To someone who has no doubt that the properties
>of this construct are objective (even if only partially determinable
>by us) the matter is unproblematic. Others have to live with the
>uncertainty that is with us in most aspects of the human condition.
There seems to be some tension between this remark and Martin's
invocation of Goedel at the end of his posting. Surely we have
additional reason for accepting ZFC, namely that the reasonings it
embodies have been used in mathematics for over a hundred years (i.e.
from some time before ZFC was formulated) and have not led to any
contradiction or inconsistency with otherwise established
mathematical results. The "paradoxes of set theory" resulted from
modes of reasoning that could not be reproduced in ZFC, at least so
far as anyone has been able to see in a very long time. One can say
that ZFC has been amply tested by "mathematical experience".
One might point out that this testing had been going on for some
years before the "intuitive Cantorian picture of sets of sets of sets
of ..." was clearly formulated as the standard interpretation of ZFC.
Regarding Martin's "honor roll" (augmented by Allen Hazen), it's
worth observing that the inconsistencies all came to light in a
relatively short time. For example Frege's Grundgesetze, volume I,
appeared in 1893, and Russell discovered the inconsistency of the
system in 1901.
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