On Lindenbaum's extensions. (Part A.)

Abstract

An extension version of this abstract will appear in Reports on Mathematical Logic. 1. Consider the well-known Lindenbaum lemma: If X is a consistent set of formulas then there exist consistent and complete set Y such that X C Y. It is fairly obvious that the Lindenbaum's extension Y is not uniquely determined (cf. [1], [2], [3], [4], [6]). However Tarski has proved in [8], a theorem concerning the power of the class of Lindenbaum's extensions: (1.1) (Tarski): If {p (q p),p [(p q) q], (q s) [(p q) (p s)]} = A and A C X then the only Cn* - consistent and Cn* - complete extension of the consistent set X is the class of all two-valued implicational tautologies (Z2). (Cn* is the consequence operation based only on the modus ponens and substitution rules). Later this Tarski's problem was considered in regard to another systems (cf. [2], [6], [7]). In the present paper we shall prove, among others, the existence of the weakest propositional calculus for which the class of all two-valued tautologies is the only consistent and complete extension and we shall generalize the above mentioned Tarski's theorem. Problems considered in this paper were formulated by Professor W. A. Pogorzelski. Let S be a set of well-formed formulas built by means of propositional variables and some of the connectives + , *, =, ~. R is a set of rules of inference, Cn(R, X) is the standard consequence-operation. R0* denotes the set {r0, r*} (r0 - the modus ponens rule, r* - the substitution rule), 0 is the empty set, Z2 is the set of all two-valued tautologies (Fragment tekstu).