arXiv subjects

Ju. T. Lisica

Publications and source records attributed to Ju. T. Lisica.

3 recordsLinked to original sources

arXiv2025

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

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., $X=...\{3,\{2,\{1,\{0\}\}\}\}...\,\,$.

math.LO↗

arXiv2024

On all numbers great and small (Topological fields of Conway's numbers and their completions)

The proper Class $\bf{No}$ of all Conway's numbers $\cite{l3}$ is considered as a region of investigation. It turns out to be a total ordered Field (i.e., a field whose domain is a proper Class) and this totally, or linear ordered Class, containing the real numbers ${\mathbb R}$ and the ordinal numbers {\bf On}. For any subfield $F$ of $\bf{No}$, i.e., $F$ is a set nor proper class, considered with topology induced by a linear ordering on $F$ a completion $\tilde F$ is constructed; in particular, for $ζ=ω^{ω^μ}$, $0\leqμ<Ω$, and for a specially defined subfield $F={\mathbb P}_ζ\subset{\bf No}$ a complete subfield ${\mathbb R}_ζ\subset{\bf No}$ is defined as $\tilde {\mathbb P}_ζ$. Fundamental (Cauchy) sequences $(x_α)_{0\leqα<ζ}$ are considered in a subfield $F\subset {\mathbb P}_ζ\subset{\bf No}$, where $ζ$ is the smallest ordinal number which does not belong to $F$, and they are the main instrument in the paper. A fragment of Mathematical Analysis in ${\mathbb R}_ζ$ is given and two of its non-trivial results are presented: every positive number $x\in{\mathbb R}_ζ$ has a unique $n$-th root in ${\mathbb R}_ζ$, for each positive integer $n$ and every odd-degree polynomial with coefficients in ${\mathbb R}_ζ$ has a root in ${\mathbb R}_ζ$. Hence so-called fundamental theorem of algebra: the ring ${\mathbb R}_ζ[i]\stackrel{def}{=}{\mathbb C}_ζ$ of all numbers of the form $x+iy$ ($x,y\in{\mathbb R}_ζ$), $i^2=-1$, is an algebraically closed field.

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arXiv2012

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

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.

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