We introduce a new notion of SLDNF-tree and prove a completeness result for SLDNF-resolution that extends a theorem by Stä rk: not only the existence of some successful (finitely failed, resp.) SLDNF-tree is established, but it is shown that every fair SLDNF-tree is successful (finitely failed, resp.).

By [12] we know that transfinite induction up to ΘεΩN+10 is not provable in IDN, the theory of N-times iterated inductive definitions. In this paper we will show that conversely transfinite induction up to any ordinal less than ΘεΩN+10 is provable in IDNi, the intuitionistic version of IDN, and extend this result to theories for transfinitely iterated inductive definitions. In [14] Schütte proves the wellordering of his notational systems using predicates is wellordered) with Mκ ≔ {x ∈ and 0 ≤ κ ≤ N. Obviously the predicates are definable in IDNi with the defining axioms: where Prog [Mκ, X] means that X is progressive with respect to Mκ, i.e. The crucial point in Schütte's wellordering proof is Lemma 19 [14, p. 130] which can be modified to where TI[Mκ + 1, a] is the scheme of transfinite induction over Mκ + 1 up to a.

The main features of the IBM Electronic Data Processing Machine, Type 702, are discussed from the programroar's point of view to illustrate how it was designed specifically to solve large accounting and statistical problems in business, industry, and government.The 702 exploits in one integrated system the high speed and storage capacity of magnetic tape, the 'accessibility of electrostatic memory supplemented by large auxiliary storage on magnetic drums, the flexibility of punched-card document input, the page printing output of modern accounting machines, and the technology of general-purpose, stored-program, electronic computer s.The 702 is a serial machine with decimal arithmetic.Its serial nature provides several unusual logical features of great aid in programming accounting problems.