[FOM] truth and consistency

[Martin Davis]

At 10:16 AM 6/1/2003, Charles Parsons wrote:
>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.

Charles is right that I overstated the case, but in my defense I should 
remark that the great bulk of ordinary mathematics uses only a very limited 
part of ZFC, in fact a good deal less even than the original Zermelo 
axioms. Tony Martin's proof of determinacy  for Borel sets is notable for 
actually requiring omega levels of the hierarchy. So classical mathematics 
can't really be invoked in favor of the consistency of full ZF.

Martin