and S,"*. n-here systems 8: and Sf* are ext'eiisioiis of RASIOWA-SLUPECKI'S system S, and S,* (see [TJ and 181). Then we shall show that for every cardinal number In, n-here 0 < 111 5 P o , there exist a system ST,,, of propositional calculus and a system SP,,, of predicate calculus such that the syst>eni ST,,, has exactly m Lindenbaum's E( ,322) is tlie set of all valid formulas in the matrix m. By %$' we denot'e the well-know classical matrix and 11-22 we denote t'he set of all two-valued t)autologies from tlie set A' ' . Hence E(91;) = 2 ; . The cardinal degree of completeness of a propositional calculus A' we denote syml,olically by d(S) and by y ( S ) n-e denot,e the ordinal degree of completeness of tlie calculus S (cf. [lo). pp. 100-101). tenis and the system SP,,, has exact)ly nt Lindenbaum's oversystems. D e f i n i t i o n 1 .
.,'R. X) E ('pl" e (Va
E A' " -Ch(R; I)) Ch(X: X u [&I) = S". c'R. S> E Cpl" o (Vp E S A -Cn(R. X)) YJ E Cn(R: X u (9)). D e f i n i t i o n 9 . D e f i n i t i o n 3.
( R , X ) E SCpl e (VIZ E 5"") ( V p E IS") (Ve: d t + S") ([h'(n)
, Y u (a}) = 8 ' 1 . 1 D e f i n i t i o n 5 . Lv(Cn(R, X ) ) = ( Y S": y 4 Y 2 Cn(R, 9 ) A A (VP, E S" -Cn(R, Y)) y E Cn(R, Y u (m}) A Cn(R,