Quantum Mechanics and the Continuum Problem (with Referee Reports)

It is shown that Feynman's formulation of quantum mechanics can be reproduced as a description of the set of intermediate cardinality. Properties of the set follow directly from the independence of the continuum hypothesis. The concept of discrete space is always regarded as a unique alternative to continuous space. Nevertheless, there is one more possibility originated from the continuum problem: since discrete space is a countable set, the set of intermediate cardinality represents the golden mean between the opposing concepts. But although the continuum problem has been solved [1], the status of the set of intermediate cardinality is still unclear. The commonly held view is that, according to the independence of the continuum hypothesis (CH), we can neither prove nor refute existence of the intermediate set and, therefore, CH or its negation must be taken as an additional axiom of standard Zermelo-Fraenkel set theory (ZF). Thus we get two alternative set theories: with and without the set of intermediate cardinality. The choice is a matter of experimental verification analogously to the choice of true geometry caused by the independence of fifth Euclid's postulate. In other words, the independence of CH converts existence of the intermediate set into a physical problem of determination of the set-theoretic structure of space-time. Assume that the set of intermediate cardinality exists and consider the maps of the intermediate set I to the sets of real numbers R and natural numbers N .