 By Andrzej Mostowski

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Lecture XIII. ' .. Lecture XIV. Theory of models for non-elementary languages . .. Lecture XV. Problems in the foundations of set theory Lecture XVI. On direct and reduced products . . . . . . . . .. Bibliography Index of authors Index of subjects 5 7 10 18 27 34 43 51 62 71 81 90 99 107 119 132 140 150 151 172 174 Foreword In the summer of 1964 I delivered in the Summer School in Vaasa, Finland, a series of lectures on the development of mathematical logic and of the study of foundations of mathematics in the years 1930 -196~.

Let us assume that to each integer n there is associated a numeral Ii and that the formula 171 =1= Ii is provable in T whenever m and n are different integers. We shall say that a formula F with one free variable is a weak description of a set X of integers if for any integer n the formula F(fi) is provable in T just in case n is an element of X. Sets which possess at least one weak description in T are called weakly representable in T. There are only denumerably many such sets because the number of different formulae is denumerable.

Same relationship exists between certain axiomatic theories based on the classical and the intuitionistic logic. g. Peano's arithmetic based on the classical logic is interpretable in Peano's arithmetic based on the intuitionistic logic. This theorem gives us an intuitionistic consistency proof for Peano's arithmetic based on the classical logic. It is remarkable that this proof should turn out to be so easy while no strictly finitistic consistency proof exists. Peano's arithmetic based on the intuitionistic logic contains thus many non-Iinitistic elements.