Now, the mentioned widely accepted claim, supported by books of famous physicists, [10], that “the quanta violate the Bell Inequalities”- a claim to which recently the further claim was added in [6–8] that the last loopholes in its experimental verification were at long last closed - amounts in the above formulation to the existence of a mathematical theorem, let us denote it by *S*, such that
$$\begin{array}{}S\in {\mathcal{T}}_{q}\end{array}$$(5)

In other words, there exists a theorem *S* among the set 𝓣_{q} of theorems of the above mentioned relevant part of the theory of quanta, such that
$$\begin{array}{}S\u27f9(non\phantom{\rule{thinmathspace}{0ex}}{T}_{Bell})\end{array}$$(6)

Let us now recall that it is a typical and general feature of all sets of theorems 𝓣 in every axiomatic mathematical theory to have the following implication hold
$$\begin{array}{}({S}^{\prime}\in \mathcal{T},\phantom{\rule{thinmathspace}{0ex}}{S}^{\u2033}\in \mathcal{F},\phantom{\rule{thinmathspace}{0ex}}{S}^{\prime}\u27f9{S}^{\u2033})\u27f9{S}^{\u2033}\in \mathcal{T}\end{array}$$(7)

where, with the above notation, 𝓕 is the set of all wff-s of the respective axiomatic theory, [11]. Similarly, we have as a typical and general feature of all sets of wff-s 𝓕 in every axiomatic mathematical theory the implication
$$\begin{array}{}{S}^{\prime}\in \mathcal{F}\u27f9non\phantom{\rule{thinmathspace}{0ex}}{S}^{\prime}\in \mathcal{F}\end{array}$$(8)

And then (5) - (7) give
$$\begin{array}{}non\phantom{\rule{thinmathspace}{0ex}}{T}_{Bell}\in {\mathcal{T}}_{q}\u2acb{\mathcal{T}}_{set}\end{array}$$(9)

which together with (3), (8) implies
$$\begin{array}{}{T}_{Bell},\phantom{\rule{thinmathspace}{0ex}}non\phantom{\rule{thinmathspace}{0ex}}{T}_{Bell}\in {\mathcal{T}}_{set}\end{array}$$(10)

that is, 𝓣_{set}, and thus Set Theory, is contradictory, since it contains *both* the theorem *T*_{Bell} *and* its negation *non* *T*_{Bell}.

However, the set 𝓣_{set} of all mathematical theorems of Set Theory is considered to be *contradiction free*, although so far that property - or for that matter, the negation of that property - has *not* yet been proved.

Lastly, let us show that given any *S* in (5), then we have the following logical consequence, and at the same time, also a much *stronger* version of (6), namely
$$\begin{array}{}[S\u27f9(non\phantom{\rule{thinmathspace}{0ex}}{T}_{Bell})]\u27f9[\mathrm{\forall}T\in {\mathcal{T}}_{set}:(S\u27f9(non\phantom{\rule{thinmathspace}{0ex}}T)]\end{array}$$(11)

or in other words, the relation 6 holds not only in the *particular* case of the Bell inequalities given by the theorem *T*_{Bell}, but also for *all* valid mathematical statements in Set Theory, that is, for *all* theorems *T* ∈ 𝓣_{set}.

In particular, as mentioned in [11, 12], it is true that:

If a valid quantum statement - be it of theoretical or experimental nature, yet expressed mathematically by *S* ∈ 𝓣_{q} - violates the Bell Inequalities, then that statement *S* must also violate *all* other valid mathematical statements *T* ∈ 𝓣_{set}, and thus in particular, relations such as 0 = 0, 1 = 1, 2 = 2, 3 = 3, ….

Indeed, the proof of (11) is as follows. Let us assume that (11) does not hold, and let us reformulate it conveniently in a logically equivalent form. For that purpose, we recall that, according to basic and elementary rules of the binary valued Logic of Aristotel - Chrysippus, we have the following *logical equivalences*, denoted by ≡, for arbitrary wff-s *Q*, *Q*′, Q″ ∈ 𝓕
$$\begin{array}{rcl}non\phantom{\rule{thinmathspace}{0ex}}(non\phantom{\rule{thinmathspace}{0ex}}Q)\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}& \equiv & \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}Q\\ (Q\u27f9{Q}^{\prime})\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}& \equiv & \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}(non\phantom{\rule{thinmathspace}{0ex}}Q)\vee {Q}^{\prime}\\ non\phantom{\rule{thinmathspace}{0ex}}(Q\u27f9{Q}^{\prime})\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}& \equiv & \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}Q\wedge (non\phantom{\rule{thinmathspace}{0ex}}{Q}^{\prime})\end{array}$$

hence regarding (11), we have
$$\begin{array}{rcl}(S\u27f9(non\phantom{\rule{thinmathspace}{0ex}}{T}_{Bell}))\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}& \equiv & \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}(non\phantom{\rule{thinmathspace}{0ex}}S)\vee (non\phantom{\rule{thinmathspace}{0ex}}{T}_{Bell})\\ (non\phantom{\rule{thinmathspace}{0ex}}(S\u27f9(non\phantom{\rule{thinmathspace}{0ex}}{T}_{Bell})))\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}& \equiv & \phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}(S\wedge {T}_{Bell})\end{array}$$

thus the expression in (11) is logically equivalent with
$$\begin{array}{}(S\wedge {T}_{Bell})\vee (\mathrm{\forall}T\in {\mathcal{T}}_{set}:(S\u27f9(non\phantom{\rule{thinmathspace}{0ex}}T))\end{array}$$

And then assuming that (11) does not hold, we obtain
$$\begin{array}{}((non\phantom{\rule{thinmathspace}{0ex}}S)\vee (non\phantom{\rule{thinmathspace}{0ex}}{T}_{Bell}))\wedge (\mathrm{\exists}{T}_{0}\in {\mathcal{T}}_{set}:S\wedge {T}_{0})\end{array}$$(12)

In this way, both statements ((*non S*) ∨ (*non T*_{Bell})) and (∃*T*_{0} ∈ 𝓣_{set} : *S* ∧ *T*_{0}) must be valid. And since in view of (5), *S* was supposed to be valid, it follows that non *T*_{Bell} must also be valid. And then, we are again back to the contradiction in (10).

Thus (11) does indeed hold.

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