Skands and coskands (The non-founded set theory with individuals and its model in the Field of all Conway numbers)

TL;DR

This paper introduces new set-theoretic concepts called skands and coskands within a modified NBG framework, exploring non-founded sets, and models them using Conway numbers, challenging traditional axioms like regularity.

Contribution

It develops the theories of skands and coskands, extending NBG set theory by relaxing the regularity axiom and incorporating infinite-length structures and Conway numbers.

Findings

Introduces the concept of skands as decreasing tuples of founded sets.

Defines coskands as increasing tuples of founded sets.

Models these structures within the field of Conway numbers.

Abstract

The basic one in this work is the axiomatic set theory $NBG$ (von Neumann-Bernays-G{\"o}del), which is a first-order theory with its own axioms, including in particular the axiom of choice ${\bf AC}$ and the axiom of regularity ${\bf RA}$. The universal class ${\bf V}$ of all sets in this theory exactly coincides with the class of all founded sets, i.e., such $X\in{\bf V}$ that {\it does not exist} an infinitely descending $\in$-sequence $X\ni X_1\ni X_2\ni...\ni X_n\ni...$ of sets $X_n$, $n=1,2,3,...\,\,$. In the first part of the paper, a new concept of {\it skand} is introduced -- a random aggregate, or \grqq decreasing\grqq\, tuple composed of founded sets, e.g., $X=\{1,\{2,\{3,\{...\,\,\,...\}\}\}\}$, and the theory of $NBG^-=NBG-{\bf RA}$, i.e., the theory of $NBG$ without the axiom of regularity ${\bf RA}$, to which is added the new axiom ${\bf SEA}$ of the existence of infinite-length skands and the pseudo-founding axiom ${\bf PFA}$. These new axioms are a negation of the axiom of regularity and are thus less restrictive than the axiom of regularity ${\bf RA}$ in the sense that they admit the existence of non-founded sets, and the axiom of regularity excludes the existence of such sets. At the same time of course the axiom of extensionality ${\bf EA}$ is replaced by a more accurate axiom of extensionality ${\bf EEA}$, since it takes into account the equality of new objects. In the second part of the paper, a new concept of {\it coskand} is introduced, which is dual to a notion of skand and is a random aggregate, or \grqq increasing\grqq\, tuple composed of founded sets and the theory of $NBG$ and actually is a theory $NBG[\cal U]$ with individuals as limiting coskands, e.g., $X=...\{3,\{2,\{1,\{0\}\}\}\}...\,\,$.