[FOM] Questions on Cantor

[Vaughan Pratt]

(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

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

and

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