[FOM] extramathematical notions and the CH
JoeShipman at aol.com
Wed Jan 30 08:13:53 EST 2013
It's possible we could use physics to gain access to noncomputable objects, if an experimentally measurable dimensionless physical quantity is a noncomputable real number; in that case ZFC would only decide finitely many bits of the number and if we could measure more than that we would have new mathematical knowledge coming from physics.
My point was that such knowledge would be about absolute statements, but it could still go beyond computability even if it would never settle CH.
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On Jan 30, 2013, at 4:48 AM, Sam Sanders <sasander at cage.ugent.be> wrote:
> Physics may have much to teach us about math, but in my opinion it will have nothing to say about non-absolute math like CH. I would be very interested in criticisms of this opinion.
I agree with Joe Shipman. For instance, given that we can only measure up to finite precision, how would we even measure
if a given object is computable or non-computable?
Even if there were to be non-computable objects "out there", we would run into problems of the following nature:
Harrington's theorem (WKL_0 is \Pi_1^1-conservative over RCA_0) implies that there is both a recursive and non-recursive model for \Pi_1^1-statements provable in WKL_0. One can then argue that physics does not really involve even \Pi_2^1-statements, as "there exists a function" is actually "there exists a function up to finite precision", and the latter is an arithmetical quantifier.
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