[FOM] Weak foundations for cohomological number theory

[Vaughan Pratt]

On 1/4/2011 4:40 AM, Harvey Friedman wrote:
> A good next step would be to do this in full ZFC with Power Set
> weakened to
>
> 100 power sets of omega exists.
>
> Then reduce 100 considerably, maybe down to 0.

Yes!  I like that!

Except that I would go for one power set of omega rather than none, with 
the caveat that the power set of omega should be understood not really 
as a set but as a complete atomic Boolean algebra.  You ok with that, 
Harvey?

The "power set" of that particular CABA, defined as the complete 
homomorphisms from it to 2, is in fact a set, and it is omega.  From 
this point of view taking the "power set" of *anything*, however 
structured, is an involution.

In this way you have a set-sized algebra closed under the "power set" 
operation.  Z, ZF, and ZFC can only offer a class-sized algebra 
(internally speaking) when closed under that operation (unless Harvey 
has something up his sleeve there).

Why classes are considered a virtue of Z(F(C)) is something I would love 
to have explained to me.

What must be given up here is the idea that every set is discrete. 
Instead every "set" X should be considered a pair (X, K^X) where K^X 
consists of the maps from X to K.  One should spend a year or so getting 
used to this idea by taking K = 2 (as far as Barwise and Seligman took 
the idea), and then move on to larger K as illustrated for K = 4 at 
http://boole.stanford.edu/pub/bhub.pdf (the paper behind a talk I'll be 
giving in Bhubaneswar next month).  For quantum mechanics, statistics, 
etc. K can be the complex rationals.

To get even further than that requires 
http://boole.stanford.edu/pub/CommunesFundInf2010.pdf , which has just 
appeared in

http://www.mimuw.edu.pl/~fundam/FI/previous/vol103.html

(Volume 103 of Fundamenta Informaticae).  This extends the above basic 
idea to incorporate sorts, along with properties as dual to sorts.  It 
pushes both the philosophy of mathematics and the mathematics of 
philosophy into hitherto unexplored areas, including places where a 
coherent notion of C.I. Lewis's quale (pl. qualia) can be found.  Inter 
alia this provides Edmond Wright's very recent (2008) collection "The 
Case for Qualia" with an internally consistent logical basis.

(Gosh, I was able to say all that without the bonus phrase "Chu space." 
  Very important stepping stone, Chu spaces.)

Vaughan Pratt