By Andrzej Mostowski (Eds.)

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54 (1964), pp, 291-302. [G] F. Gal v i n, Horn sentences, Ann. Math. Logic 1 (1970), pp, 389-422. [Kl] H. J. K e i s 1e c, Model theory for infinitary logic, Amsterdam 1971. [K2] H. J, K e i s I e r, Logic with the quantifier "there exist uncountably many", Ann. Math. Logic 1 (1970), pp, 1-93. [L] L. Lip n e r, Some aspects of generalized quantifiers, Ph. D. Thesis, University of California, Berkeley 1970. [U] P. Lin d s t rom. First order predicate logic with generalized quantifiers, Theoria 32 (1966), pp.

Lecture II The incompleteness of arithmetic In this lecture we shall be concerned with another important contribution of the early thirties to the study of the founda tions of mathematics, especially with the so called first incompleteness- theorem of Godel [54] which states that the usual axiomatic systems of arithmetic of integers are incomplete. In order to explain the, importance of this result we insert some brief historical comments. Since the publication of the works of Frege and of Russell and Whitehead logicians believed that each intuitively correct deduction can be reconstructed in the classical logical calculus.

More precisely, if a theory T has infinite models and