von Neumann’s Theorem Revisited

Hilbert-style axiomatic completion: On von Neumann and hidden variables in quantum mechanics

“…Thus, von Neumann had clearly accomplished his mathematical goal of demonstrating that ordinary probability theory (Probability) (in conjunction with (Quantities) and (Incompatibility)) was "sufficient" for the "unambiguous derivation" of the trace rule. Yet one point should be emphasized here, following Acuña: the immediate goal of his "Aufbau" was to identify the basic probabilistic-theoretical assumptions sufficient for deriving the trace rule Acuña [2021a] Acuña [2021b]. 20 As such, A.…”

“…However, in the main, I do not disagree either on the historical, physical, or mathematical facts. Indeed, excellent exegetical work has already been done on von Neumann's work in physics Janssen, 2013] [Lacki, 2000] [Rédei, 1996] [Rédei, 2006] [Rédei and Stöltzner, 2006] [Stöltzner, 2001] [ Bueno, 2016], including on his no hidden variables proof [Bub, 2011] [Bub, 2010] [Dieks, 2017] [ Mermin and Schack, 2018] [Stöltzner, 1999] [Acuña, 2021a] [Acuña, 2021b]. Instead, my disagreement concerns primarily the framing, which lumps von Neumann in with Bohm and Bell (especially the latter).…”

Logical Characterization of Contextual Hidden-Variable Theories based on Quantum Set Theory

“…However, the algebraic structure of beables does not directly treat the logical structure of observational propositions nor the structure of the language speaking of beables, so that we do not have a formal framework to treat, for instance, Bohr's original notion of the "classical language" to describe beables in a given measurement context, or Hardy's logical formulation of non-locality. 1 Here, we introduce a new approach based on quantum set theory to provide a logical framework for modal interpretations. Quantum set theory was introduced by Takeuti [37] for constructing mathematics based on quantum logic and developed by the present author [26][27][28][29][30][31][32]; the relationship with topos quantum theory was studied by Eva [15] and Döring et al [13].…”

“…The first general proof of the impossibility theorem for non-contextual hidden-variable theories was given by von Neumann[25]; see[1,10,11,24] for the recent debate on the status of von Neumann's impossibility proof. Later, Kochen and Specker[22] proved the theorem for the Hilbert space with the dimension greater than 2 under the sole requirement that hidden-variables satisfy functional relations for observables; a similar result can be derived as a corollary of Gleason's theorem[16] 4.…”

Einstein Completeness as Categoricity

“…See[13] for a discussion of completeness as understood by Russell and Whitehead, and an argument that Gödel's own understanding of completeness, as negation-completeness, is relevantly different than the descriptive variety 4. See[14] for a recent re-evaluation of (the debate on) von Neumann's proof of completeness 5. Perhaps the most detailed formal reconstruction of the EPR argument has been given in McGrath[15] where the following is also noted: "Regrettably EPR equate two notions of completeness: 'complete representation by a wave function' and 'complete theory' are used interchangeably."…”