By Herbert B. Enderton

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It will be our aim to study these results so as to obtain a picture of how the philosophical programs of the three major schools have influenced the formal development of logic and of the foundational study. We will see that the contribution of each of them has been great and that none of them could exist without the others. The three schools underwent great changes during the years 1930 ~ 1960. Especially striking has been the development of metamathematics which originally aimed at a proof of consistency as envisaged by Hilbert but which has later developed into a much mo~e ambitious theory.

These were the well-known works by Heyting [80). Godel [54] and Tarski [222]. We shall begin our exposition by discussing these papers arrd certain other works which immediately depend on them. Lecture I Formalization of the intuitionistic logic Intuitionism as invented by Brouwer rests on several general principles, only some of which are relevant to intuitionistic logic. Very important but not relevant to our immediate purpose is the assumption that general set-theoretical notions are not to be admitted into mathematics and that all mathematics is to be reduced to the arithmetic' of integers and to a very special intuitionistic theory of the continuum.

B aid win and A. H. Lac h 1a n, On strongly minimal sets, J. Symb. Logic 36 (1971), pp. 79-96. [CK] C. C. C han g and H. J. K e is I e r, Model theory, Amsterdam 1973. [FV] S. Fe fer man and R. L. V aug h t, The first order properties of products of algebraic systems, Fund. Math. 47 (1959), pp. 57-103. [Fr] H. F r i e d man, Beth's theorem in cardinality logics, Israel J. of Math. 14 (1973), pp, 205-212. [F] G. F u h r ken, Skolem-type normal forms for first order languages wIth a generalized quantifier, Fund.

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