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 $\mathbf{AC}$ and the axiom of regularity $\mathbf{RA}$. The universal class $\mathbf{V}$ of all sets in this theory exactly coincides with the class of all founded sets, i.e., such $X\in \mathbf{V}$ 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 $NB{G}^{-}=NBG-\mathbf{RA}$, i.e., the theory of $NBG$ without the axiom of regularity $\mathbf{RA}$, to which is added the new axiom $\mathbf{SEA}$ of the existence of infinite-length skands and the pseudo-founding axiom $\mathbf{PFA}$. These new axioms are a negation of the axiom of regularity and are thus less restrictive than the axiom of regularity $\mathbf{RA}$ 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 $\mathbf{EA}$ is replaced by a more accurate axiom of extensionality $\mathbf{EEA}$, 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 $NBG[\mathcal{U}]$ with individuals as limiting coskands, e.g., $X=...\{3,\{2,\{1,\{0\}\}\}\}...$.