Abstract
Quantum mechanics (QM) predicts probabilities on the fundamental level which are, via Born probability law, connected to the formal randomness of infinite sequences of QM outcomes. Recently it has been shown that QM is algorithmic 1random in the sense of Martin–Löf. We extend this result and demonstrate that QM is algorithmic \(\omega\)random and generic, precisely as described by the ’miniaturisation’ of the Solovay forcing to arithmetic. This is extended further to the result that QM becomes Zermelo–Fraenkel Solovay random on infinitedimensional Hilbert spaces. Moreover, it is more likely that there exists a standard transitive ZFC model M, where QM is expressed in reality, than in the universe V of sets. Then every generic quantum measurement adds to M the infinite sequence, i.e. random real \(r\in 2^{\omega }\), and the model undergoes random forcing extensions M[r]. The entire process of forcing becomes the structural ingredient of QM and parallels similar constructions applied to spacetime in the quantum limit, therefore showing the structural resemblance of both in this limit. We discuss several questions regarding measurability and possible practical applications of the extended Solovay randomness of QM. The method applied is the formalization based on models of ZFC; however, this is particularly wellsuited technique to recognising randomness questions of QM. When one works in a constant model of ZFC or in axiomatic ZFC itself, the issues considered here remain hidden to a great extent.
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Notes
We thank the Reviewer of FS for turning our attention to this scenario.
References
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Appendices
Appendix 1: Arithmetic Hierarchy

(1)
An arbitrary set A is in \(\sum ^0_0\) (\(\prod ^0_0\)) if the characteristic function of A is computable by a Turing machine.

(2)
2. \(A\in \sum ^0_n\), \(n\ge 1\), if there exists a computable relation \(R(x,y_1,y_2,\ldots ,y_n)\) such that
$$\begin{aligned} x\in A \Longleftrightarrow \exists _{y_1}\forall _{y_2}\exists _{y_3}\ldots \exists _{y_n,(n\text { is odd})}(\forall _{y_n, (n\text { is even})})R(x,y_1,y_2,\ldots ,y_n) \end{aligned}$$where the last quantifier is \(\exists\) (an existential one) whenever n is odd or \(\forall\) (an universal one) if n is even. Similarly \(A\in \prod ^0_n,n\ge 1\) when
$$\begin{aligned} x\in A \Longleftrightarrow \forall _{y_1}\exists _{y_2}\forall _{y_3}\ldots \forall _{y_n, (n \text { is odd})}(\exists _{y_n, (n\text { is even})})R(x,y_1,y_2,\ldots ,y_n)\,. \end{aligned}$$Now the last quantifier remains \(\exists\) if n is even and \(\forall\) otherwise.

(3)
If \(A\in \sum ^0_n\cap \prod ^0_n\) then \(A\in \varDelta ^0_n\).

(4)
If \(A\in \bigcup _n(\sum ^0_n \cup \prod ^0_n)\) then A is arithmetical.
Given an infinite binary sequence \(\sigma \in 2^{\omega }\) and the product measure \(\mu\) on the Cantor space \(2^{\omega }\), the randomness of \(\sigma\) can be defined as (Martin–Löf e.g. Downey and Hirschfeldt 2010):
1. ML test: A sequence \(\{A_n, n\in \mathbb {N}\}\) of uniformly computably enumerable (c.e.) [i.e. c.e. together with the set of its indices (Downey and Hirschfeldt 2010, p. 11)] of \({\sum }^0_1\) classes (\({\sum }^0_1\) subsets of sequences from \(2^{\omega }\)) such that \(\forall _{n\in \mathbb {N}}\,\left( \mu (A_n)<2^{n}\right)\).
2. \(A\subset 2^{\omega }\) is MLnull when there exists a ML test \(\{A_n, n\in \mathbb {N}\}\) such that \(A\subseteq \bigcap _{n\in \mathbb {N}}A_n\).
3. \(\sigma \in 2^{\omega }\) is MLrandom if \(\{\sigma \}\) is not MLnull (for each ML test).
4. A ML test \(\{A_n,n\in \mathbb {N}\}\) is universal when \(\bigcap _{n\in \mathbb {N}}B_n \subset \bigcap _{n\in \mathbb {N}}A_n\) for all MLtests \(\{B_n,n\in \mathbb {N}\}\).
Lemma 15
There exists a universal ML test.
MLrandom sequence \(\sigma \in 2^{\omega }\) is known to be 1random. The following modification of the above definition of ML randomness explains the hierarchy of nrandom sets, for all \(n\ge 1\).
i. \(\text {ML}_{n}\) test: A sequence \(\{A_k,\ k\in \mathbb {N}\}\) of uniformly c.e. of \({\sum }^0_n\) classes (\({\sum }^0_n\) subsets of sequences from \(2^{\omega }\)) such that \(\forall _{k\in \mathbb {N}}\,\left( \mu (A_k)<2^{k}\right)\).
ii. \(A\subset 2^{\omega }\) is \(\text {ML}_{n}\)null when there exists a \(\text {ML}_{n}\) test \(\{A_k,\ k\in \mathbb {N}\}\) such that \(A\subseteq \bigcap _{k\in \mathbb {N}}A_k\).
iii. \(\sigma \in 2^{\omega }\) is nrandom if \(\{\sigma \}\) is not \(\text {ML}_{n}\)null (for each \(\text {ML}_{n}\) test).
Appendix 2: Borel Sets, Infinite Binary Sequences and Polish Spaces
Borel sets BOR(X) on a topological space X comprises the smallest \(\sigma\)algebra containing all open subsets of X. (A \(\sigma\)algebra of subsets of X is a family \({{\mathcal {F}}}\) of subsets of X (\({{\mathcal {F}}}\subset {{\mathcal {P}}}(X)\)) such that (1) \(\emptyset , X\in {{\mathcal {F}}}\); (2) if \(A\in {{\mathcal {F}}}\) then \(X\setminus A\in {{\mathcal {F}}}\); (3) if \(A_n\in {{\mathcal {F}}}, n\in \mathbb {N}\) then \(\bigcup _{n\in \mathbb {N}}A_n\in {{\mathcal {F}}}\).)
A function \(\mu :BOR(X)\rightarrow [0,1]\subset \mathbb {R}\) is a (probability) measure if (1) \(\mu (\emptyset )=0, \mu (X)=1\); (2) if \(A_n\in BOR(X), n\in \mathbb {N}\) are Borel sets which are pairwise disjoint, i.e. \(\forall _{i\ne j, i,j\in \mathbb {N}}\,\left( A_i\cap A_j=\emptyset \right)\), then \(\mu (\bigcup _{n\in \mathbb {N}})=\sum _{n\in \mathbb {N}}\mu (A_n)\); we also require \(\mu\) to be nonatomic i.e. (4) for every set \(A\in BOR(X)\) with \(\mu (A)>0\) there exists \(B\in BOR(X)\) such that \(B\subset A\) and \(0<\mu (B)<\mu (A)\). If X is a topological group, e.g. in the case \(X=(\mathbb {R},+)\), the translational invariance is additionally required: (5) \(\mu (A)=\mu (A+t)\) for all \(A\in BOR(X)\) and \(t\in X\). An example for such a measure is the Lebesgue measure on \(\mathbb {R}\) i.e. on \(BOR(\mathbb {R})\). We call a subset \(A\subset X\) to be measurable if there exists a Borel set \(B\in BOR(X)\) such that \(A\triangle B :=(A\setminus B)\cup (B\setminus A)\) is a set of measure zero, i.e. \(\mu (A\triangle B)=0\).
Mathematicians found a suitable representation for \((\mathbb {R}, BOR(\mathbb {R}))\) by another topological spaces and their Borel subsets, that have simplified proofs and have allowed to grasp important properties of subsets of \(\mathbb {R}\). This is precisely analogous to set theory, where working in different models of ZF(C) has exhibited deep invariant properties of the line of real numbers (real line) along with its regularity properties (e.g. Bartoszyński & Judah, 1995; Jech, 2003). Choosing suitable model of ZF(C) leads to in a sense minimal representations for \(\mathbb {R}\), where important and demanded properties of reals are highlighted and formally grasped. At the same time one does not loose any important information about objects under studies. Such possibility is based on Borel isomorphism of topological spaces. Given two topological spaces X and Y, we say they are Borel isomorphic if there exists a function \(f:X\rightarrow Y\) such that for each Borel set \(B\in BOR(Y)\) the set \(f^{1}(B)\) is Borel in X, i.e. \(f^{1}(B)\in BOR(X)\). The Borel isomorphism between spaces allows to work with any such space when dealing with regularity properties, such as Lebesgue measurability, Baire property, measure zero sets and others, due to the following
Lemma 16
Let \(f:X\rightarrow Y\) be a Borel isomorphism of X and Y. A set A is Lebesgue measurable in X (of first category, with Baire property, measure zero) \(\iff\) f(A) is Lebesgue measurable (of first category, with Baire property, measure zero) in Y.
In fact, all Polish spaces (i.e. topological spaces homeomorphic to a complete metric space with no isolated points) are pairwise Borel isomorphic up to certain measure zero sets (Bartoszyński and Judah 1995, Th. 1.3.19). An example of a Polish space could be the real line \(\mathbb {R}\), and to replace it by another Polish space, we need a rather strict Borel isomorphism of \(\mathbb {R}\).
Appendix 3: Generic Ultrafilters in Models of ZFC
One defines topology on a poset by taking open sets to be \(O \subseteq P\), for which if \(p\in O\), \(q\in P\) and \(q\leqslant p\), then \(q\in O\). A subset \(D\subseteq P\) is dense if for every \(p\in P\) there exists \(d \in D\) such that \(d\leqslant p\). A filter on \(\langle P,\leqslant \rangle\) is a subset \(F\subset P\) which is (1) nonempty; (2) If \(p\in F\) and \(q \in F\), then there is \(r\in F\) such that \(r\leqslant p\) and \(r\leqslant q\); (3) if \(p\in F\), \(q\in P\) and \(p\leqslant q\), then \(q\in F\). Two elements (called conditions) \(p,q\in P\) of a partial order P are compatible, if there exists \(r\in P\) such that \(r\leqslant p \wedge r\leqslant q\). Otherwise we say p, q are incompatible. An antichain in P is a set of pairwise incompatible elements. Partial order \(\langle P, \leqslant \rangle\) fulfills the countable chain condition (CCC) if every antichain in P is countable.
Recall that \(\aleph _0\) is the cardinality (cardinal number) of \(\omega\) (or \(\mathbb {N}\)). Then the forcing axiom (Martin axiom) states that for a partial order P satisfying the countable chain condition and for any family \({{\mathcal {D}}}\subset {{\mathcal {P}}}(P)\) of fewer than \(2^{\aleph _0}\) of dense subsets of P, there exists a filter \(G\subset P\) such that \(G\cap D\ne \emptyset\) for all \(D\in {{\mathcal {D}}}\) (here \({{\mathcal {P}}}(P)\) is the power set of P). The forcing axiom has to be assumed for higher cardinalities, although for countable families of dense subsets it holds
Lemma 17
For any partially ordered set \(\langle P, \leqslant \rangle\) and for a family \({{\mathcal {D}}}\) of countably many dense subsets of P and \(p\in P\), there exists a filter G on P such that \(p\in G\) and \(G\cap D \ne \emptyset\) for every \(D\in {{\mathcal {D}}}\).
Such filters G are called generic ultrafilters and their existence is crucial for forcing procedure.
Let us see that Cohen forcing is of this type. The elements of P are finite \(\{0,1 \}\)sequences, hence they are the members of \(2^{<\omega }\). The condition p is stronger than the condition q (written \(p<q\)), if p extends q. Let M be a ground model, \(\langle P,\leqslant \rangle \in M\) and let \(G\subset P\) be generic over M (G intersects all subsets of P that are dense in M. For every \(n\in \mathbb {N}\) the sets \(D_n=\{p\in P\,\, n\in \mathrm {dom}(p) \}\) (p is a finite sequence, thus a function, and \(\mathrm {dom}(p)\) is its domain) are dense in P and hence they intersect G. Crucially, the function \(\bigcup G\) is now in \(2^{\omega }\) which is not in M (Theorem 2) and it is a characteristic function of a certain subset of \(\mathbb {N}\). Thus it is represented by an infinite \(\{0,1\}\)sequence, hence a real number r. This r is contained in M[G], although it is not present in M, and it is precisely a Cohen generic real.
Appendix 4: BooleanValued Models and Forcing Extensions
A model \(M^B\) is constructed in M, where B is a complete Boolean algebra B in the model M. Let \({{\mathcal {U}}}\) be an ultrafilter in B and let \(P\subset B\) be a partial order completed to B; thus \({{\mathcal {U}}}\) is also a generic ultrafilter G in P (in M). The model \(M^B\) assigns Boolean values \(\Vert \phi (x)\Vert \in B\) to formulas of the ZFC language interpreted in \(M^B\). Given the ultrafilter G, one defines the equivalence relation \(\equiv _G\) on the Bterms \(x,y\in M^B\) by
so that the corresponding \(\in\)relation E on equivalence classes \([x], [y]\in M^B/G\) emerges
Measure Algebra as Maximal Boolean Subalgebra of Projections in \(\mathbb {L}(\mathcal {H})\)
Theorem 9
(Kadison & Ringrose, 1997, Th. 9.4.1) Let \(\dim {\mathcal {H}}=\infty\), let \(\mu\) be the Lebesgue measure on the \(\sigma\)algebra of Borel subsets on [0, 1] and let \(\mathcal {L}(\mathcal {H})\) be the extension of the space of bounded linear operators on \(\mathcal {H}\) containing also unbounded operators. Maximal abelian von Neumann subalgebras (MASA’s) of \(\mathcal {L}(\mathcal {H})\) are unitarily equivalent to
Thus maximal complete Boolean subalgebras in \(\mathbb {L}\) (for \(\dim {\mathcal {H}}=\infty\)) have the general form of
Lemma 18
\(B= B_a\oplus B_c\), where \(B_a\) is the atomic Boolean algebra and \(B_c\) is the measure algebra B.
B is the algebra of Borel subsets of [0, 1] modulo the ideal \(\mathcal {N}\) of \(\mu\)measure zero Borel subsets
Lemma 19
(de Groote, 2005; Kadison & Ringrose, 1997) B is atomless Boolean algebra.
The above property really distinguishes infinitedimensional case, since
Corollary 3
(de Groote, 2005; Kadison & Ringrose, 1997) If \(\dim {\mathcal {H}} < \infty\) then maximal complete Boolean algebras of projections chosen from the lattice \(\mathbb {L}\) are atomic.
Due to the atomless property of B the nontrivial random forcing is an inherent part of the QM formalism (e.g. Jech, 1986, Proposition 2.1). This is also responsible for some random Solovay generic phenomena in QM and helps understanding the enhancement of random infinite binary sequences on infinitedimensional Hilbert spaces of states. In terms of measurement procedure in Sect. 3.1 the binary random sequences r which are irreducible to sequences in M come from random extensions M[r] and the measurement procedures accessible in M. Namely \(r\notin M\) and the relation \(V^B/U_r=M[r]\) holds (\(V^B\) is the Booleanvalued model of ZFC and M[r] is the 2valued model) such that r represents the result of quantum measurement performed on infinitedimensional Hilbert space in M[r]. This Hilbert space in M[r] is not any tensor product of finitedimensional state spaces in M (cf. also the Theorem 5.1 in Landsman, 2020).
Appendix 5: QM is 1Random
The method of showing that QM is random in a strict mathematical sense relies on formalization as defined in Król and AsselmeyerMaluga (2020) and follows the argumentation that QM is 1random from Landsman (2020) which we present here briefly.
Let A be a selfadjoint operator in the space \(\mathcal {B}(\mathcal {H})\) of bounded operators on \(\mathcal {H}\). The extension of the Born measure \(\mu _A\) over \(2^{\omega }\) follows the general procedure of extending a measure \(\mu\) on a measurable space \((X,\mu )\) onto the measure \(\mu ^{\infty }\) on \(X^{\omega }\). At first one considers cylindrical \(\sigma\)algebra S of subsets of \(X^{\omega }\) generated by sets of the form
where \(X_k=X\) for all \(k\ge N+1\) and \(C_i\in Bor(X)\) for all \(i=1,2,\ldots ,N\). Then the measure on all sets \(\mathbf {C}\in S\) is given by
The probability measure \(\mu ^{\infty }_S\) on S (\(\mu (X)=1\)) extends uniquely to the probability measure \(\mu ^{\infty }\) on \(X^{\omega }\), which is essentially the \(N\rightarrow \infty\) case of Eq. (23). The uniqueness follows from Dudley (2002, Th. 8.2.2).
Thus given the Born measure \(\mu _A\) on the spectrum \(\sigma _A\) of a selfadjoint operator A in the state \(\psi\), we are looking for the induced measure on \(\sigma _A^{\omega }\) where infinite sequences of the outcomes of repeating measurements of A in the state \(\psi\) live. This is essentially \(\mu ^{\infty }=\mu ^{\infty }_A\) on \(X=\sigma _A\) generated by \(N\rightarrow \infty\) limit of Eqs. (24) and (23) as discussed above.
Let us consider now \(X=2=\{0,1\}\) and take the product measure \(\mu ^{\infty }\) on \(2^{\omega }\). The infinite binary sequence of QM outcomes can be obtained by repeating measurements of an elementary projection P in a state \(\psi\), where the process is suitably prepared by making reset on a system every time each measurement has been performed (e.g. Benioff, 1976a). Note that \(\sigma _P=\{0,1\}\) for any projection P. Also in this case the Born measure \(\mu _P\) determines uniquely the measure \(\mu _P^{\infty }\) on \(2^{\omega }\) and this elementary case is sufficient to capture randomness of binary sequences in QM. Namely it holds (Landsman, 2020, Th. 4.1, Cor. 5.2)
Lemma 20
According to the measure \(\mu _P^{\infty }\), the set of 1random sequences in \(2^{\omega }\) has measure 1. The measure of the set of nonrandom binary infinite sequences generated by the Born measure \(\mu _P\) in QM vanishes.
It follows that the Born’s probability of occurrence of infinite binary sequence of outcomes in QM which is nonrandom is zero. Given the definition of nrandomness, [nRAND], of \({\text {QM}^{\star }}\) and hence QM in Sect. 3.2, we conclude that
Corollary 4
QM is 1random.
This is extended to nrandomness via Solovay generic forcing extensions assigned inherently to QM in the main body of the paper (see Sect. 3.2).
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Król, J., Bielas, K. & AsselmeyerMaluga, T. Random World and Quantum Mechanics. Found Sci (2022). https://doi.org/10.1007/s10699022098522
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DOI: https://doi.org/10.1007/s10699022098522
Keywords
 Quantum mechanics
 Randomness
 Model theory
 Forcing