and Vn>n.(fj(n) I~ Fi(n)), it follows that Ls-- I ~ and Ls--1 >Fj. On the basis of Theo- rem i, taking the relation /~0Wj into account, we can conclude that there exists a k~ 0WiC~A such that F k = L s -- i. Since L~%n [max(n q-l,F~(n) q-l)], it follows that F~ Fi. Consequently, fi~ ~a and i ~ ~A (f). The theorem is proved. We get the following corollaries from Theorem 4 and the Remark. COROLLARY i. Each infinite recursive set has infimum. COROLLARY 2. There exists a recursively enumerable nonrecursive set that has infimum. It has been shown in [7] that each infinite recursively enumerable set can be decom- posed into two disjoint infinite nonspeedable sets. Therefore, the following corollary holds. COROLLARY 3. Each infinite recursively enumerable set can be decomposed into two dis- joint infinite sets, each of which has infimum. The following problem is still unsolved: Does there exist an infinite recursively enum- erable set that does not have infimum? 1. 2. 3. 4. 5. 6. 7. LITERATURE CITED P. R. Young, "Toward a theory of enumerations," J. Assoc. Computo Mach., 16, 328-348 (1969). P. R. Young, "Speed-ups by changing the order in which sets are enumerated," Math. Sys- tems Theory, 5, 148-156 (1971). L. A. Levin, "Step-counting functions of computable functions," Supplement i, in: Com- plexity of Computations and Algorithms [in Russian], Mir, Moscow (1974), pp. 174-185. H. Rogers, Jr., Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York (1967). B. A. Trakhtenbrot, Complexity of Algorithms and Computations [in Russian], Novosibirsk State Univ. (1967). M. Blum and I. Marques, "On complexity properties of recursively enumerable sets," J. Symbolic Logic, 38, No. 4, 579-593 (1973). I. Marques, "On degrees of unsolvability and complexity properties," J. Symbolic Logic, 40, No. 4, 529-540 (1975). THEORY OF ZE~LO WITHOUT POWER SET AXIOM AND THE THEORY OF ZERMELO--FRENKEL WITHOUT POIFER SET AXIOM ARE RELATIVELY CONSISTENT V. G. Kanovei Introduction. The theory ZF of Zermelo--Frenkel without the axiom of choice is consis- tent with the theory Z of Zermelo without the axiom of choice because the totality of all sets of rank <m -~ is a model of Z in ZF [i]. In view of GDdel's second theorem, the theories ZF and Z are therefore not relatively consistent (assuming, of course, the consis- tency of the theory ZF). The content of the present paper is the proof of the following theo- rem, which shows that the situation is different if we remove the power set axiom from the theories under consideration: THEOREM. The theories Z- and ZF-, obtained from Z and ZF, respectively, by omitting the power set axiom, are relatively consistent. Moreover, ZF- has an interpretation in Z-. The consistency of %- and ZF- relative to each other (and for a series of other theo- ries) was announced without proof in [2] with reference to [3], where an equivalent result is stated without proof. The author does not know if proofs of these theorems have been published. Moscow Institute of Rail Transport Engineers. Translated from Matematicheskie Zametki, Vol. 30, No. 3, pp. 407-419, September, 1981. Original article submitted July 25, 1978. 0001-4346/81/3034-0695507.50 9 1982 Plenum Publishing Corporation 695