FIN-SET: A SYNTACTICAL DEFINITION OF FINITE SETS

Slavisa B. Presic

Abstract: We state Fin-set, by which one founds the notion of finite sets in a syntactical way. Any finite set $\{a_1,a_2,\dots,a_n\}$ is defined as a well formed term of the form $S(a_1+(a_2+(\cdots+(a_{n-1}+a_n)\cdots)))$, where $+$ is a binary and $S$ a unary operational symbol. Related to the operational symbol $+$ the term-substitutions (1) are supposed. Definition of finite sets is called syntactical because by two algorithms Set-alg and Calc (below) one can effectively establish whether any given set-terms are equal or not equal.

All other notions of finite sets, like $\in$, ordered pair, Cartesian product, relation, function, cardinal number are defined as a corresponding term. Each of these definitions is recursive. For instance, $\in$ is defined by \begin{align*} &x\in S(a_1)\quad\text{iff}\quad x=a_1
&x\in S(a_1+\cdots+a_n)\quad\text{iff}\quad x=a_1 \text{ or } x\in S(a_2+\cdots+a_n)
&x\notin\emptyset\quad (\emptyset\text{ denotes the empty set}) \end{align*}

Keywords: finite set, algorithm, syntactical definition

Classification (MSC2000): 03E30

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