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Da Costa and E. H. Alves. que poun le ealeul C1 , C. R. Acad. Sc. Paris 283 A, pp. 729-731 . 1977. <. Cn' Formal Logic XVIII, Notre Dame Journal of pp. 621-630. N. C. A. da Costa and L. Dubikajtis. 1968. '<', Bull. Acad. Polonaise des Sciences XVI, pp. 551-557. 1977. in Non-Classical Logics, Model Theory and Computability (Eds, A. I. Arruda, N. C. A. da Costa and R. Chuaqui), North-Holland, Amsterdam, pp. 37-56. e, N. C. A. da Costa and M. Guillaume. Sun lu ealeul-6 Cn, Anais da Academia Brasileira de Ciencias, 36, pp.

Lc, to appear. 197+c. t>, to appear. A. 1. Arruda and N. C. A. da Costa. lJrlm, c. R. Acad. Sc. Paris 259, pp, 1964. 2943-2945. L, Boletim da Sociedade Matematica de Sao Paulo 18, fascs. 19 e 29, pp. 83-89. 1968a. 6epMation, Notices AMS 15, pp. 399-400. 1968b. 6epMation, Notices AMS 15', p. 555. 1970. {on, Nagoya Mathematiaal Jurnal 38, pp. 7184. 1974. 6 ]n' Mathematica Japonicae 19, pp, 183-186. 1965. 30 1977. AYDA I. ARRUDA Une 4emantique po«4 te catQUt C~, C. R. Acad. Sc. Paris 284 A, pp. 279- 282.

2 (ii) of H with a O = an and the sequences and (3 p-q " to conclude that indeed r v y < (3 p-q ) t/J" (y). Take x , E V ~ a. ~ a" The last rule of inference is "from t/J(v) -+ ¢ infer 3yt/J(y) -+ ¢" and checking the validity of the claim for this rule is quite similar to the previous Case. ON STRONG AXIOMS OF INDUCTION 53 T of quantifier rank < k are in T k, ~Je shall assume that k is large enough (;;, 10) so that all the axioms but comprehension and replacement could be in Tk, We pass over extensionality, the existence of the empty set, foundation, and the axiom of choice as these follow easily.