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

Abstract

The basic one in this work is the axiomatic set theory $NBG$ (von Neumann-Bernays-G{ö}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., $X=...\{3,\{2,\{1,\{0\}\}\}\}...\,\,$.