Some theorems on the smallest sets closed under the classes of relations and their generalizations. I

Abstract

The concept of a smallest set including an initial set and closed under the relations (in particular: operations) of a given class is of a considerable importance in the metho? dology of deductive sciences. The sets of significant expressions, the sets of theses of axiomatic systems, the sets of consequences of some sets of propositions are examples of the smallest sets including certain initial sets and closed under definite operations. In the methodology of deductive sciences we intuitively use various theorems on the smallest sets. The proofs of some theorems of such a kind are given in ? 1 of this article. These proofs are formalized (by means of the method presented in the book: J. Slupecki and L. Borkowski, Elements of Mathematical Logic and Set Theory). This formalization may be useful, on the one hand, in the applications of such theorems in teaching mathematical logic; on the other hand, it makes easy stating which of these theorems are valid for some generalizations of the concept of a smallest set in? cluding an initial set and closed under some relations. Such generalizations are also suggested by some examples used in the methodology of deductive sciences. If we say e.g. about expressions formed of the infinite number of symbols or if we say about the rule of infinite induction (co-rule), then we take into consideration operations which assign some expressions to infinite sequences of expressions or to infinite sets of ex? pressions. Some theorems in which we use a suitably generalized concept of a smallest set are given in ? 2 and ? 3 of this article.