On a Formalization of Cantor Set Theory for Natural Models of the Physical Phenomena

This article presents a set theory which is an extension of $ZFC$. In contrast to $ZFC$, a new theory admits absolutely non-denumerable sets. It is feasible that a symbiosis of the proposed theory and Vdovin set theory will permit to formulate a (presumably) non-contradictory axiomatic set theory which will represent the core of Cantor set theory in a maximally full manner as to the essence and the contents of the latter. This is possible due to the fact that the generalized principle of choice and the generalized continuum hypothesis are proved in Vdovin theory. The theory, being more complete than $ZF$ and more natural according to Cantor, will allow to construct and study (in its framework) only natural models of the real physical phenomena.