[FOM] Truth and Consistency

[Lucas Wiman]

Charles Parsons writes:

 >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".

While I agree that this is a good reason to keep using ZFC, I don't 
think that this tells us that ZFC is consistent.  This tells us that if 
ZFC has any inconsistencies, then they are probably extremely 
non-obvious.  They might have been missed by mathematicians and set 
theorists for so long because they're extremely weird or long or 
something like that.  This is thus a contingent fact about human 
psychology.

Michael Dummett has the following to say on induction of this sort:

"Why, then, does [Hartry Field] believe ZF to be consistent?  ...  The 
reason offered by Field himself for believing in the consistency of ZF 
is that `if it weren't consistent someone would have probably discovered 
an inconsistency in it by now'; he refers to this as inductive 
knowledge.  To have an inductive basis for the conviction, it is not 
enough to observe that some theories have been discovered to be 
inconsistent in a relatively short time; it would be necessary also to 
know, of some theories not discovered to be inconsistent within 
three-quarters of a century, that are consistent.  Without non-inductive 
knowledge of the consistency of some comparable mathematical theories 
there can be no inductive knowledge of the consistency of any 
mathematical theory." ("What is Mathematics About?", Mathematics and 
Mind, Alexander George, ed., Oxford: 1994)

I am uncertain whether this is correct; we have no non-inductive 
verification of the facts of physics, yet they are generally considered 
inductively verified.  If we take Popper's falsification view, then 
clearly it's quite reasonable to conjecture that ZF is consistent since 
so little disproof exists of it, though accepting it as a verified fact 
may be too strong.  Like in physics, this must remain tentative.  
Physicists discovered experiments which showed that Newtonian mechanics 
is false, though many (most?) of its core assumptions were preserved.  
At that time, it was an extremely successful theory which had managed to 
explain nearly every phenomenon of celestial mechanics.  I think the 
situation is similar to ZF.  It's extremely well-confirmed in one sense, 
but there are likely areas in which it is not. However, the amount of 
work that has been done in ZF tends to suggest that if any 
inconsistencies are found in ZF, then they will likely be at the extreme 
fringes of "human" mathematics (as the Burali-Forti and Russell 
paradoxes were).

- Lucas Wiman