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

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).