SOME SET THEORIES ARE MORE EQUAL ... - Logic at Harvard

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10 MENACHEM MAGIDORproperty , so if we consider different axioms that could be supportedby the previous principles, then having generic absoluteness providesadditional appeal to the axiom.If possible the axiom should be resilient under forcingextensions.5. Can Set Theory be relevant to Physics?In the previous section we presented the process of search for new axiomswhich is not dissimilar to the formation of theories in any branchof science. Can we carry this analogy even further and the decide betweendifferent set theories on the basis on their scientific consequences,say in Physics?As to be expected we do not have any definite case in which differentset theories have an impact on physical theories but we believe thatthe possibility that may happen in the future is not as outrageous asit may sounds . Here are two examples which relates to basic issues inQuantum Mechanics. Two famous arguments against hidden variableinterpretation of Quantum Mechanics are the Bell Inequalities ([4]. Seealso [38]) and the Kochen-Specker Theorem. ([19]).Bell inequalities imply for instance that for spin 1 particles , the2correlation which is observed between the measurements of the spinof two entangled particles along different axis can not realized by afunction that assigns to any direction in space a definite value which isthe value of the spin along the given direction. (We identify directionsin space with points on the the two dimensional unit sphere S 2 . ) .A natural requirements form the function (which does not exist!) isthat if will be measurable with respect to the usual measure on S 2 .Pitowsky in a series of papers ([32],[33],[34]) showed that if one allowsa larger class of functions then one can have a deterministic modelrealizing correlations which violates the Bell inequalities . (The classof functions Pitowsky considered were still nice enough so that we canstill talk about integration, correlation etc. ) But in order to showthat a spin function realizing the Quantum Mechanical statistics heassumed CH or MA (Martin’s axiom). The use of additional axiomsis necessary in view of the following result:Theorem 5.1 (Farah,M.[7]). If the continuum is real valued measurablethen Pitowski’s kind spin function does not exists. The same holds
SOME SET THEORIES ARE MORE EQUAL 11in the model one gets from any universe of ZFC by adding (2 ℵ 0) + randomreals. 4Similar situation exists with respect to the Kochen-Specker theorem.It deals with spin 1 particle. It follows from Quantum Mechanics thatfor such a particle and given three mutually orthogonal directions inspace ⃗ X, ⃗ Y , ⃗ Z , while one can not measure simultaneously the valueof the spin along this three axis ,(The corresponding operators do notcommute), one can still measure simultaneously the absolute value ofthe spin along the three given axis and then the three values we getare always two 1’s and one 0’s. The Kochen Specker theorem claimsthat such behavior can not be realized by a deterministic function. i.e.there is no function F on S 2 which gets the values 0, 1 which satisfythe conditions:(1) For all ⃗ X ∈ S 2 F (− ⃗ X) = F ( ⃗ X)(2) For every triple of mutually orthogonal vectors in S 2 : ⃗ X, ⃗ Y , ⃗ ZF ( ⃗ X) + F ( ⃗ Y ) + F ( ⃗ Z) = 2.(The theorem is really a finitary statements and hence there are norestriction on the function F ).Call such a function A KS function.Pitowsky in [34] suggested relaxing the requirements above on thefunction F so that clause (2) is assumed to hold almost always in thefollowing sense : For a fixed vector ⃗ X ∈ S 2 the set of ⃗ Y such that if ⃗ Zthe a vector orthogonal to ⃗ X and ⃗ Y then F ( ⃗ X) + F ( ⃗ Y ) + F ( ⃗ Z) ≠ 2is of measure 0 with respect to the usual measure on the great circleperpendicular to ⃗ X. As before Pitowsky shows that under Martin’saxiom MA there does exists a function on S 2 satisfying the modifiedrequirements. Call such a function a PKS function. Similarly to thelast theorem we are able to proveTheorem 5.2 (Farah,M.). If the continuum is real valued measurablethen there is no PKS function on S 2 does not exists.The same holds inthe model one gets from any universe of ZFC by adding (2 ℵ 0) + randomreals.The functions that Pitowsky constructs above are not measurableand so probably will rejected as having physical meaning in the samesense that the partition of the sphere given by the Banach-Tarski paradoxlacks physical meaning , but the Pitowski functions are not aspathological and there is some variation of measure theory in whichthese functions are rather well behaving. As Pitowsky says in [32]:4 Shipman in [39] announced that the same conclusion is consistent with ZFC.We were not able to reconstruct his arguments neither directly nor after correspondingwith him).
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