SKAND THEORY AND ITS APPLICATIONS. (A NEW LOOK AT NON-WELL-FOUNDED SETS)

Abstract

A new mathematical object called a skand is introduced, which turns out in general to be a non-well-founded set. Skands of finite lengths are ordinary well-founded sets, and skands of very long length (like the hyper-skand of all ordinals) are hyper-classes.

Self-similar skands are also considered, and they clarify the reflexivity of sets, i.e., the meaning of the relation \(X \in X\); in particular, self-similar skands considered as non-well-founded sets are always reflexive, but not vice versa. The existence of self-similar skands shows at once that all the well-known set-theoretical paradoxes are not paradoxes at all, and hence are not necessarily fatal for any set theory. E.g., the inconsistency of Russell’s “set” \(R=\{X\mid X\notin X\}\) is proved here not with the help of Russell’s paradox (as it is traditionally given, which is incorrect), but via a simple method of the maximality (universality) of \(R\) which goes back to Cantor and is also applied to other set-theoretical paradoxes.

Generalized skands are also defined and a new look at the generalized skand-class of all ordinals is demonstrated. In particular, the last (class) ordinal called the eschaton is defined.

The next application of skand theory is a description of all epsilon-numbers in the sense of Cantor. Another application is a generalized theory of one-dimensional continua of arbitrary powers and the construction of generalized real numbers as a non-Archimedean straight line of arbitrary power, and the introduction of the absolute continuum and the absolute straight line as the hyper-classes nearest to the class of sets.