Abstract

According to a well-known of A. Lindenbaum every consistent set of sen- tences of an arbitrary deductive theory can be extended to a consistent and complete system. The question arises how many such extensions there are, in other words, how great the number of all consistent and complete systems which include a given set of sentences is. By Ro-systems we mean systems based on modus ponens and sub- stitution rule and by Ro-systems we mean systems based only on modus ponens. We say that a system has Tarski's property of power , where is an arbitrary cardinal number such that > 0, if the cardinality of all Lindenbaum's extensions of this consistent system is . The content of this paper is the following: First, we prove that there does not exist a minimal Ro-system with Tarski's property of power , where 0 < < 2 @0 , and that there exists a minimal Ro-system with Tarski's property of power of the continuum. Then we shall prove that for every cardinal number there exists a consistent Ro- system including all two-valued tautologies , which has Lindenbaum's extensions. We would like to notice here that T. Stepien in [5] proved that there does not exist a minimal Ro-system with Tarski's property of power for 0 < < 2 @0 . Independently, A. Biela solved the above problem for = 1 and proved that for any Ro-system with Tarski's property of powerLindenbaum's Extensions 43 1 there exists a family of descending Ro-systems with Tarski's property which has an empty intersection of the set of theses. We use the standard logical notation. Let S be the smallest set containing the set At = fpi : i 2 N g and closed under joining of formulas by means of the connectives of the set . We assume in this paper that the considered systems hR; Xi full the following condition: for every Y S such that Cn = S , there exists a nite set Z Y for which Cn = S . By L we denote the set of all Lindenbaum's extensions of the set Cn. The symbol hR; Xi hR0 ; X0 i means that the system hR; Xi is a subsystem of the system. The relation is an equivalence of the systems. We introduce the following order relation: jhR; Xij jhR0 ; X0 ij i hR; Xi is a subsystem of the system hR0 ; X0 i. By L() we denote the length of the formula . Card means the cardinal number of the set X. For p 2 At, by S p we denote the set of all formulas built of only one propositional variable p