[FOM] Arithmetical soundness of ZFC (platonic)

[Nick Nielsen]

On Wed, Jul 29, 2009 at 7:54 AM, Timothy Y. Chow <tchow at alum.mit.edu> wrote:

>As for Weaver's suggested dichotomy between "anything goes" and "be
>absolutely safe," Zermelo (as I suggested) did not fall into either of
>these camps:

> In solving the problem we must, on the one hand, restrict these
> principles far enough to exclude all contradictions and, on the
> other hand, leave them sufficiently wide to retain all that is
> valuable in this discipline.

Probably a thinker of Zermelo's stature could not be easily
pigeon-holed, but clearly the two provisions in the above quote
suggest the same two camps: the "anything goes" camp (vaguely
platonist) places the emphasis on "sufficient width" while the "be
absolutely safe" camp (vaguely constructivist, and which we might want
to call the "safety first" camp) places the emphasis upon "far enough
to exclude all contradictions". How sufficient is "sufficient"? How
far is "far enough"? Any sensible doctrine is going to try to balance
opposed movements of thought, as in the Zermelo quote; in so far as a
movement of thought promotes one uncompromising doctrine over another,
it tends to ideology rather than reason. Reasonablenss is all about
compromise, and it is a compromise that Zermelo recommends in the
above quote.

Best wishes,

Nick Nielsen