[FOM] Choice of new axioms 1
joeshipman@aol.com
joeshipman at aol.com
Tue Feb 14 20:34:11 EST 2006
Friedman:
In my response to Shipman I said that very weak fragments of second
order
arithmetic are much more than sufficient to handle all of the standard
physical theories. This is very well known.
Shipman:
You are again talking about logical strength after coding, not ontology.
Friedman:
There are a number of conservative extensions of very weak fragments of
Z_2
that avoid this coding, and that are totally satisfactory. All of this
is
well known. E.g., people calling themselves predicativists (and others)
know
this very well, and they can respond to this point if they wish.
It is more delicate to give such conservative extensions of EFA =
exponential function arithmetic, a very weak fragment of PA of special
importance for f.o.m. But there is no doubt that this can be done fully
satisfactorily.
Shipman:
The fact that these conservative extensions that avoid coding have sets
of reals and sets of sets of reals in them is what is relevant to my
point, not that they are conservative over a weak fragment of Z_2.
To actually DO PHYSICS, one needs to be able to talk about physical
objects, and in these theories, physical objects correspond to
operators on Hilbert space, or connections on Riemannian manifolds, or
other objects of high logical type. When you code into second order
arithmetic, all the usefulness of the theory to physicists disappears.
If you DON'T code and instead use one of these conservative extensions,
you are still using objects of high logical type, even if you have
restricted yourself so that you won't be able to prove certain
statements of lower type.
To truly avoid real numbers, it would not even be enough to code things
into second-order arithmetic in such a way that standard physical
concepts retained their existence in the theory, which is the minimum
requirement for a theory to be acceptable from the physicist's point of
view. Second-order arithmetic still essentially involves infinite
objects, and "arbitrary precision" or "infinite dvisibility". You could
code things further, and speak only about algorithms and experimental
setups, and abandon "physical objects" altogether, but what remains
hardly deserves the title of "physical theory"; I think it makes more
sense to take the current "best theories" at face value.
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