Quantum mechanics and the Continuum Problem

In one-dimensional case, it is shown that the basic principles of quantum mechanics are properties of the set of intermediate cardinality. PACS numbers: 03.65.Bz, 02.10.Cz The concept of discrete space is not a unique alternative of the continuous space. Since discrete space is a countable set, there is an intermediate possibility connected with the continuum problem: space may be neither continuous nor discrete. The commonly held view is that the independence of the continuum hypothesis (CH) is not a certain solution of the continuum problem in consequence of incompleteness of set theory. Nevertheless, from the independence of CH follows a unique definite status of the set of intermediate cardinality. It is important here that this set must be a subset of continuum (continuum must contain a subset equivalent to the intermediate set). Taking into account that any separation of the subset is a proof of existence of the intermediate set, which contradicts the independence of CH, we get that the set of intermediate cardinality exists only as a subset of continuum. In other words, the subset of intermediate cardinality, in principle, cannot be separated from continuum (set theory “confinement”). If Zermelo-Fraenkel set theory is consistent, complete, and giving the correct description of the notion of set, then this is the only possible understanding of the independence of CH. Note that if we postulate existence of the intermediate set (in other words, if we take the negation of CH as an axiom), the result will be the same: since any construction or separation of the set are forbidden by the independence of CH, we have to reconcile with the “latent” intermediate subset in continuum. ∗e-mail: [email protected]