Re: Continuum hypothesis

aatu.koskensilta@xxxxxxxxx says...

The result the original poster probably has in mind is that both ZFC
+CH and ZFC+~CH are conservative over ZFC for arithmetical statements,
i.e. statements in which the quantifiers range over the hereditarily
finite sets (these are equivalent to statements in the first-order
language of arithmetic, of course). The conservativity extends to
Pi^1_4 sentences (in the arithmetical hierarchy), if I recall
correctly.

One would think that such results would imply that the continuum
hypothesis has no significant consequences for applied mathematics.
That's probably true, but there is at least one application of the
continuum hypothesis to a problem arising in theoretical physics.

If you prepare an electron with its "spin" aligned with an axis
S1 (that is, the projection of its spin on the axis S1 is +1/2),
and then immediately afterwards measure its spin along a different
axis S2, the results are probabilistic: If theta is the angle between S1
and S2, then:

No matter what direction S2 you choose, there are only two possible
outcomes for a measurement of spin along the direction S2: +1/2
and -1/2. With probability cos^2(theta/2), you will measure +1/2,
and with probability sin^2(theta/2), you will measure -1/2.

The question arises as to whether there is a "hidden variables"
explanation for this result. Such an explanation could, for
instance, take the form of a function

f : DxD -> {+1/2, -1/2}

where D is the set of possible directions of an axis upon
which to project the spin. Then f(S1,S2) will yield +1/2 or
-1/2 to give the result of measuring the spin in direction
S2 given that it was prepared with spin in direction S1.
To agree with the predictions of quantum mechanics, it must
be that for every S1 and for all angles theta,

m({ S2 | f(S1,S2) = +1/2 and the angle between S1 and S2 = theta })
= cos^2(theta/2)

where m() is the Lebesgue measure for the set of possible
directions S2 (which can be identified with the points on a
unit sphere).

Bell's Theorem proves that no measurable function f can possible
satisfy this constraint. However, Pitowsky proved that if one
assumes the continuum hypothesis, one can construct a nonmeasurable
function that satisfies this constraint.

http://edelstein.huji.ac.il/staff/pitowsky/papers/Paper%2001.pdf

It's hard to believe that nature would actually take advantage
of nonmeasurable functions and the continuum hypothesis, but it's
an interesting result.

--
Daryl McCullough
Ithaca, NY

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