Skand theory and its applications. (A new look at non-well-founded sets)

TL;DR

This paper introduces skand theory, a new mathematical framework for non-well-founded sets, offering fresh insights into set reflexivity, paradoxes, and generalized continuum concepts.

Contribution

It develops the concept of skands, clarifies set reflexivity, resolves Russell's paradox via maximality, and constructs generalized real numbers and continua.

Findings

Skands include well-founded sets and hyper-classes.

Self-similar skands are always reflexive.

Russell's paradox is addressed through maximality, not contradiction.

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 is a member of 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 Russell's well-known paradox is not a paradox at all. The inconsistency of Russell's "set" R, which is the collection of all sets that are not members of themselves, 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 can be 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.