April 20
Michał Tomasz Godziszewski,
University of Warsaw
Set-theoretic independence of existence of some local hidden variable models in the foundations of quantum mechanics
In 1982 I. Pitowsky gave a construction of local hidden variable models (i.e. descirptions of quantum systems that provide deterministic predictions and are satisfied for spatially separated observables) for the so-called spin-1/2 (and spin-1) particles in quantum mechanics. Specifically, Pitowsky's main result was that under the assumption of the Continuum Hypothesis there exists a spin-1/2 function. The function constructed in the proof of the theorem is non-measurable, making Pitowsky's model not directly subject to famous Bell's theorem, stating that under certain assumptions local hidden variable models are impossible. Since the construction uses CH, the natural question is whether the existence of Pitowsky's functions is provable in ZFC.
In 2012, I. Farah and M. Magidor demonstrated that it is actually independent. Namely they proved that:
(1) if there exists a $\sigma$-additive extension of the Lebesgue measure to the power-set of the reals (i.e. if the large cardinal axiom known as 'the continuum is a real-valued measurable cardinal' holds), then Pitowsky's models do not exist, and
(2) Pitowsky's models do not exist in the random real model.
The second result thus shows that the non-existence of Pitowsky's functions is relatively consistent with ZFC. The proofs of these theorem rely on the results by H. Friedman and D.H. Fremlin, specifying that under the assumptions of (1) and (2) Pitowsky's functions have to be Borel-measurable. During the talk I will try to sketch Pitowsky's construction, explain the proofs of the Farah-Magidor theorems, and, if time permits, relate them to some ongoing debates in the quantum foundations and philosophy of set theory.