[FOM] how would a physicist know that we are not living in a Skolem hull?
Vaughan Pratt
pratt at cs.stanford.edu
Fri Feb 8 22:20:38 EST 2013
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
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