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–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 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 \(r\in 2^{\omega }\), 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 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.

量子力学 (QM) 在基本层面上预测概率，通过玻恩概率定律，这些概率与 QM 结果的无限序列的形式随机性相关联。最近已经证明 QM 是 Martin-Löf 意义上的算法 1-random。我们扩展了这个结果并证明 QM 是算法的\(\omega\) -随机和通用的，正如 Solovay 强制算术的“小型化”所描述的那样。这进一步扩展到 QM 在无限维希尔伯特空间上成为 Zermelo-Fraenkel Solovay 随机的结果。此外，更有可能存在一个标准的传递 ZFC 模型M，其中 QM 在现实中表达，而不是在集合的全域V中。然后每个通用的量子测量都会增加M无限序列，即随机实数\(r\in 2^{\omega }\)，模型经历随机强制扩展M [ r ]。整个强迫过程成为 QM 的结构成分，并与量子极限中应用于时空的类似构造平行，因此在此极限中显示出两者的结构相似性。我们讨论了关于 QM 的扩展 Solovay 随机性的可测量性和可能的​​实际应用的几个问题。采用的方法是基于ZFC模型的形式化；然而，这特别适合识别 QM 的随机性问题。当一个人在 ZFC 的常数模型或公理化 ZFC 本身中工作时，这里考虑的问题在很大程度上仍然是隐藏的。