[FOM] extramathematical notions and the CH
joeshipman at aol.com
joeshipman at aol.com
Mon Jan 28 01:26:28 EST 2013
I showed in my Ph.D. thesis (see the October 1990 Transactions of the AMS) that Pitowsky needed to assume some axiom going beyond ZFC to develop his version of quantum mechanics, because he needed incompatible measurements of non-commuting observables to correspond to changing the order of integration in an iterated integral, and I showed that "Strong Fubini Theorems" to the effect that iterated integrals (of non-negative functions) always match if they exist are consistent with ZFC.
However, Pitowsky's models are very unphysical. Rather than being realistic alternatives to standard quantum mechanics, they are existence proofs that certain philosophical interpretations of quantum mechanics are not, as had previously been argued, entailed by our current theories.
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.
-- JS
-----Original Message-----
From: Tom Dunion <tom.dunion at gmail.com>
To: fom <fom at cs.nyu.edu>
Sent: Mon, Jan 28, 2013 12:36 am
Subject: [FOM] extramathematical notions and the CH
In the "Con(ZFC) is trivial" thread (Jan. 22, 2013) Harvey Friedman said
>> As Goedel pointed out, it [Con (ZFC)] cannot be proved in ZFC, but
>>might be proved using extramathematical notions.
Sure, why not? Perhaps the CH itself might eventually be
generally accepted as true (or false) for like reasons.
As Rudy Rucker, for one, has remarked, "...it could be that one of the
reasons set theory is stalemated by the continuum
hypothesis is that we have not yet made enough attempts to identify
the problem with some problems outside of pure mathematics." (see his
1982 book Infinity and the Mind, p. 253)
In particular, physics has issues that may encroach on f.o.m.
topics.Though a FOM-like forum, the "theoretical physics stack
exchange" did not prove viable, there are yet some who seek a "realistic"
underpinning for quantum mechanics (some even to the point of
considering a logically possible "local hidden variables" theory) who look
to issues such as a non-standard theory of probability.
And in the case of the late Prof. Itamar Pitowsky (note here:
http://edelstein.huji.ac.il/staff/pitowsky/papers/Paper01.pdf) he
acknowledged explicitly his need of at least a consequence of Martin's
Axiom, if not the full strength of the CH. Perhaps some such thinkers
might be enticed to engage these types of issues at FOM.
-- TD
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