[FOM] extramathematical notions and the CH
joeshipman at aol.com
joeshipman at aol.com
Tue Feb 5 18:54:57 EST 2013
Koskensilta:
> This also answers Aatu Koskensilta's concern, because CH and V=L and
> similar non-absolute axioms cannot decide any arithmetical
> statements that were not already decidable in ZF.
Sure, but "algorithm A provides an approximation to the constant C"
is not an arithmetical statement unless C has an arithmetical
description. The (hypothetical) situation I had in mind was where C is
defined in terms of e.g. sets of reals or other more exotic
mathematical structures, and to prove the correctness of this or that
(computable or at least arithmetical) approximation we must use e.g.
CH or other "non-absolute" principles. This possibility is not ruled
out by the usual absoluteness results.
I reply:
Any constant which can be approximated arbitrarily closely by an algorithm is by definition arithmetical.
However, I can make your example work if by approximation you don't mean "arbitrarily close approximation".
If your physical theory defines a non-arithmetical real constant, there will exist some rational interval of nonzero width such that it is independent of ZFC whether the constant lies in that interval, so experiment can still give you confidence about mathematical facts that ZFC doesn't prove.
Although this is logically possible, I was basing my opinion on the fact that no known physical theory ever looked like this, or involved non-absolute mathematics in a way relevant to predicting experimental results. but all of the current theories involve quantities which can be expressed as countably infinite summations of computable numbers.-- JS
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