Proof of consistency of foundations of mathematics Alexander Kuzichev Faculty of Mechanics and Mathematics, Moscow State University, 119992, Moscow, Leninskie Gory, Russia [[email protected]] 2000 Mathematics Subject Classification. 01K Following the ideas of A.N. Kolmogorov and A.A. Markov, we propose a solution to the Hilbert problem of formally syntactically constructing provably complete and consistent foundations of mathematics in the form of calculus. The solution to the problem is derived non-axiomatically, but directly by pre- senting a two-level sequential calculus, two of whose levels adequately reflect two components of mathematics: algorithmic (calculating) and deductive (logical). On the other hand, this reflection uniquely corresponds to two principles in Cantor’s set theory distinguished to this day: classical first-order predicate logic (used by Cantor without any constraints) and Cantor’s unrestricted comprehension. The calculus M was constructed in [1], where the algorithmic component of mathematics, its reflecting first level of the calculus M, and the set-theoretic com- prehension were represented in the form of the calculus of Church’s lambda con- version [1]. The deductive aspect of each of these three parts (mathematics, the calculus M, and set theory) is represented by the postulates of the M-extension [1]. They express the classical first-order predicate logic in the sequential form of Gentzen (1934) on the basis of the lifting rule * (from the first level) with unrestricted set-theoretic comprehension in the form of lambda conversion introduced on the second level by the rules *lambda*. The completeness (with respect to Cantor’s set theory) and consistency (abso- lute consistency) of the calculus M are proved directly on the basis of the construc- tion of M and its derivations [1], [2], [3]. By virtue of Gentzen’s results (1934), the consistency of M is naturally understood and proved in the sense of Gentzen as the (absolute) consistency of the sequential construction without the cut rule. We formulated and proved the corresponding theorem on admissibility of the cut rule. In this paper (together with [1]), [2]), [3]) all constructions and proofs are obtained in the framework of Hilbert’s program. [1] Kuzichev, A.S., A Version of Formalization of Cantor’s Set Theory. Doklady Mathe- matics, Vol.60, No. 3, 1999, pp. 424-426. [2] Kuzichev, A.S., Kolmogorov Reduction and Consistency. Doklady Mathematics, Vol. 60, No. 1, 1999, pp. 32-34. [3] Kuzichev, A.S., Solution of the Hilbert Central Problem Following Kolmogorov, Vol. 61, No. 2, 2000, pp. 212-215..