From: Christian Szegedy (szegedy@or.uni-bonn.de)
Date: Mon Aug 16 2004 - 07:21:02 MDT
> I'm still learning, but it looks to me like physics describes
> relations that hold between points in quantum configuration space.
> Similarly, an axiom set, if we map it to a model, describes relations
> that hold between elements of the model. This is what suggests to me
> that the TOE of physics might be analogous to an axiomatic system.
> For me the fascinating thing is that although mathematicians can get
> along just fine using nothing but axioms, we appear to actually
> *exist* in the model, not the axioms. I am still trying to wrap my
> mind around the Lowenheim-Skolem theorem as it applies to this - would
> there be any conceivable physical test that would tell us we were
> really in a uncountable model, or would the Lowenheim-Skolem theorem
> rule that out?
Of cource, the frst step would be to define what is "physical
experiment" inside
a model (in the sense of mathematical logic). If your axiom-system is
suitably
large (that is: it allows for simulating a Turing machine), then it is
necessarily
non-complete. Even in this case, you can reduce the physically sensable
manifastation of the model by restriciting the notion of physical
experiment.
You may be able to reduce the notion of physical experiment so skilfully
that the outcome of each experiment can be decided by the axioms alone.
Of course, in this case you can tell using experiments in which model
you are.
Otherwise the theory is not complete: it does not describe the physical laws
completely.
Do you see this differently?
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