[FOM] how would a physicist know that we are not living in a Skolem hull?
meskew at math.uci.edu
meskew at math.uci.edu
Tue Feb 5 16:47:52 EST 2013
> Suppose a physicists is using a mathematical theory saying
> that there are an uncountable number of points in space.
>
> In view of Skolem's result, what would such a statement mean really?
> In the universe we are living in are there really an uncountable
> number of points in space, or is it just that reality is constructed
> so that we can never encounter the 1-1 function that counts them? In
> view of qusestions like this, what "good" are statements about the
> number of points in space to a physicist? What experiment could
> convince us about the number of points in space being
> countable/uncountable?
I hope this answer is not too glib. If the physicist believes the theory
that implies there are uncountably many points in space, then this by
itself should convince her that there are uncountably many points in
space. Now how does she defeat Skolem's paradox? Say our universe is V.
The hypothesis that in some larger universe W the set of spatial points of
V is countable, means in particular that *there is* some enumeration of
the set of spatial points of V. But the physicist's theory was not about
some "V", just "space", "functions", "points", etc. In other words there
were no restrictions on the quantifiers to some subclass; the theory just
talks about what exists, period. So the physicist just applies the
theorem that says if f is a function from natural numbers to spatial
points, then *there is* a point not in the range of f. The question, "But
is this point in our universe?" doesn't really make sense-- it exists, so
yes it exists. This is what the theory tells us.
Monroe
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