[FOM] 511: A Supernatural Consistency Proof for Mathematics

[Vaughan Pratt]

On 1/13/2013 10:12 PM, Harvey Friedman wrote:
> Con(ZFC) is not a natural principle when thinking about rainbows,
> horizons, and beauty.

Harvey, if the principle is to be "mutually interpretable with ZFC" as 
you required then I don't see how it could be inconsistent with the 
physical phenomena of rainbows and horizons, which as far as I know are 
not disputed on any logical grounds.

For beauty however I don't know where to turn but poetry.  If beauty is 
truth and vice versa then one might conclude that if Con(ZFC) is not 
beautiful then it is not true.

Four decades ago I occupied an office three doors down from Professor 
Gerry Sussman at MIT.  Gerry consistently claimed he never claimed to be 
consistent.  I was ok with at least that much since what I was doing 
didn't seem to depend on whether Gerry was consistent.  In fact my work 
turned out better when he wasn't.

Meanwhile I will continue to ponder the possibility you raise that 
consistency of ZFC is unnatural.  Maybe ZFC itself is unnatural, but in 
that case scientists as students of nature might be motivated to look 
for a logic of science, as distinct from a logic of mathematics which is 
how I've been taught to understand ZFC.

There is the further possibility that we're not even talking at cross 
purposes, which would be the exception to the usual rule.

Vaughan