[FOM] Arithmetical soundness of ZFC

[praatika@mappi.helsinki.fi]

It might be useful to split this to several distinct issues:

A potential counterexample can't be a Pi-0-1 sentence, for consistency  
amounts to Pi-0-1 soundness.

If the doubts are based on mere use of classical logic, a potential  
counterexample can't be a Pi-0-2 sentence (because the standard ZFC is  
conservative over the intuitionistic ZFC for these sentences).

If one accepts classical reasoning itself without any reservations,  
the move from the first-order arithmetic to the second-order  
arithmetic may look very natural (at least, the latter does not look  
like a random set of axioms.) The latter, however, is  
proof-theoretically equivalent to ZFC minus the power-set axiom. So  
from this perspective, at least that part of ZFC is safe, and the  
potential problems must come from the power-set axiom.

However, the power-set axiom itself is extremely natural from the  
classical point of view: after all, the standard interpretation of the  
second-order arithmetic already involves P(N), and and the step to  
P(P(N)), etc. seems then perfectly natural.

* * *

There also seem to be various different, quite natural ways of ending  
up with systems that turn out to be equivalent to ZFC (or some  
extension of it)
(Harvey, do you have some especially good examples in mind?)


* * *

But be that as it may, I would like to repeat Harvey's question: what  
would constitute a convincing proof of such arithmetical unsoundness  
of ZFC?

Of course, there are non-classical systems which do not justify  
everything that ZFC proves, but we would now need an *extremely well  
motivated* alternative non-classical foundational system which is in  
conflict with ZFC already in the arithmetical level. And that does not  
seem to be forthcoming easily.

Of course, from the classical perspective, we can speculate about the  
existence of such a counterexample (and doubt it), but from a more  
constructivistic standpoint, one needs to present a proof, and a  
concrete example.


Best, Panu



Panu Raatikainen

Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy

Department of Philosophy
University of Helsinki
Finland


E-mail: panu.raatikainen at helsinki.fi

http://www.mv.helsinki.fi/home/praatika/