[FOM] how would a physicist know that we are not living in a Skolem hull?

[Vaughan Pratt]

On 2/7/2013 4:17 PM, Dmytro Taranovsky wrote:
> At this time, we do not know whether at Planck scale, space is
> continuous or discrete (or even whether points in space are physically
> meaningful at Planck scale).

Heisenberg uncertainty requires that measuring the position of a 
particle to within distance precision x requires an uncertainty of h/x 
in its momentum (to be precise, 1/4pi of that).  Personally I interpret 
this as meaning that the "size" in SI units of the portion of the 
universe we can observe is about 1/h, or in Planck units 1/2pi.  In 
either case this is a finite quantity.

That interpretation aside, our ability to measure positions in space to 
arbitrary accuracy is limited by our ability to cope with arbitrarily 
large uncertainty in momentum.  Any particle whose momentum you cannot 
bound from above is a particle you want to stay well clear of, as you 
have no guarantee that it cannot destroy arbitrarily much of your 
neighborhood.

Hence the idea of being able to decide even whether space contains a 
countable infinity of points is already pretty far fetched.  Aleph_1 is 
Cantorianly beyond that, while c = Beth_1 is Cohenly further beyond that.

Physicists have no more insight into how to design an experiment to 
distinguish between Aleph_1 and Beth_1 than logicians have into the 
relevance of Heisenberg uncertainty to the granularity of space and 
time.  Any conference on this topic would simply have the two sides 
talking at cross purposes, much as at the Universal Algebra and Category 
Theory conference held at MSRI in July 1993 only even more so.

Vaughan Pratt