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 1-random in the sense of Martin-L¨of. We extend this result
and demonstrate that QM is algorithmic ω-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 infinite-dimensional 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 ∈ 2ω, 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 well-suited 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.