[FOM] Questions on Cantor
Frode Bjørdal
frode.bjordal at ifikk.uio.no
Wed Jan 30 11:19:24 EST 2013
Thank you very much for your useful accounts concerning well foundedness
and well orderings, Vaughan. Indeed, from an iterative conception of sets
the notions undeniably may be taken to have a conceptual priority ordering
as you suggest. For me it was not quite clear what you meant as Mirimanoff
did not prohibit what he called “ensembles extraordinaires” that are sets
that do not abide by the iterative conception.
When I wrote that EPR had been dismantled by physics I was very hesitant as
to whether I should include a modifier such as “prevailing”, but I decided
against it as I considered an impetus to a discussion concerning
foundational problems in physics to be somewhat untoward for this list. It
was nevertheless very interesting to me to read your considerations
concerning such issues. As many others, I do not find the prevailing
foundational ideas such as the Copenhagen interpretation entirely
satisfactory.
Frode Bjørdal
2013/1/29 Vaughan Pratt <pratt at cs.stanford.edu>
> (While the following strays from the original question about well-founded
> sets, this comparison between QM, that pure probability enters into
> physics, and the well-ordering theorem, that every set can be well-ordered,
> may be of independent interest to FOM.)
>
> On 1/27/2013 5:22 PM, Frode Bjørdal wrote:
>
>>
>> By the way, it took many years for the EPR paradox to be dismantled in
>> physics.
>>
>
> Certainly; in fact the "dismantling" is still on-going. Much the same can
> be said about the well-ordering theorem.
>
> Koenig's proffered refutation of the theorem was rejected only because it
> made unjustified use of a result from Bernstein's thesis three years
> earlier, pointed out by Zermelo within a day of Koenig's presentation. The
> status of the theorem itself took decades to emerge even as far as it has.
> As a proposition equivalent (in ZF at least) to Choice, we understand
> today that it is independent of ZF, although my impression is that some
> constructivists consider the well-ordering theorem false regardless of
> their opinion about Choice.
>
> Re-reading the history of the long-running Bohr-Einstein debate at
>
> http://en.wikipedia.org/wiki/**Bohr%E2%80%93Einstein_debates<http://en.wikipedia.org/wiki/Bohr%E2%80%93Einstein_debates>
>
> I see I may have misremembered which of Einstein's arguments was disposed
> of within a day when I made the comparison with Koenig's 1904 argument.
> The incident in question took place three years into the debate but five
> years before the 1935 EPR paper, namely at the 1930 Solvay meeting. There
> Einstein presented his box argument, which sought to establish the precise
> time and energy of light emitted from a spring-supported box through a
> shutter by (i) opening the shutter arbitrarily briefly to allow a photon to
> escape, (ii) adding mass m to the box to exactly compensate for the
> resulting loss of mass of the box by bringing the box back down to its
> original height, and (iii) inferring the energy as E = mc^2. Since
> (according to Einstein) the shutter speed can be arbitrarily fast and m
> measured to arbitrary accuracy, we have a violation of Heisenberg's
> uncertainty principle as applied to time and energy.
>
> The article quotes Leon Rosenfeld:
>
> "It was a real shock for Bohr...who, at first, could not think of a
> solution. For the entire evening he was extremely agitated, and he
> continued passing from one scientist to another, seeking to persuade them
> that it could not be the case, that it would have been the end of physics
> if Einstein were right; but he couldn't come up with any way to resolve the
> paradox. I will never forget the image of the two antagonists as they left
> the club: Einstein, with his tall and commanding figure, who walked
> tranquilly, with a mildly ironic smile, and Bohr who trotted along beside
> him, full of excitement...The morning after saw the triumph of Bohr."
>
> Bohr chose to attack Einstein's claim that the mass defect could be
> measured to arbitrary accuracy. This defect being h/(c*lambda) per photon
> of wavelength lambda, if the emitted photon is visible light, say with
> lambda = 500 nm, then an electron (of mass 9.11 x 10^{-31} kg) is more than
> two hundred thousand times heavier than the reduction in box weight
> Einstein is proposing to measure. Einstein would then need to propose a
> method of measuring such a tiny displacement without running afoul of
> Heisenberg uncertainty at some point in the method, a very tall order.
>
> Bohr could however have argued just as well that the shutter cannot be
> opened arbitrarily briefly if there is to be any reasonable chance of a
> photon escaping. That's simpler than going through the math of the masses
> involved.
>
> Simpler yet is to point out that classical reasoning like Einstein's is
> unsound in the quantum world. I don't know when that point of view became
> sufficiently accepted as to make it a defense against Einstein's arguments,
> but it seems pretty standard today.
>
> That doesn't mean that it has the force of logic, in fact as it stands
> it's circular. Had the EPR team thrown in the towel and turned to classical
> chaotic dynamics in competition with quantum chaotic dynamics they might
> have done better, see e.g. Arjendu Pattanayak's pages at
>
> http://www.people.carleton.**edu/~apattana/Research/**RiceTalk.html<http://www.people.carleton.edu/~apattana/Research/RiceTalk.html>
>
> and
>
> http://www.people.carleton.**edu/~apattana/Research/index.**html<http://www.people.carleton.edu/~apattana/Research/index.html>
>
> Quantum mechanics is by no means a closed book yet, and not necessarily on
> the ideological grounds that seem to differentiate the various
> interpretations of quantum mechanics. There may be more hope for this than
> finding non-ideological grounds for deciding Choice and the Well-Ordering
> Theorem.
>
>
> Vaughan Pratt
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>
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