machines and implementations The first definition of an abstract machine was given by Turing, in the classic [20]. Without repeating here the well-known definition (e.g., see [6]), 13 we recall that each Turing machine M is equipped with a "semi-infinite tape" which it uses both to compute and also to communicate with its environment: To determine the value f(n) (if any) of the partial function 14 f : N * N computed by M , we put n on the tape in some standard way, e.g., by placing n + 1 consecutive 1s at its beginning; we start the machine in some specified, initial, internal state q 0 and looking at the leftmost end of the tape; and we wait until the machine stops (if it does), at which time the value f(n) can be read off the tape, by counting the successive 1s at the left end. Turing argued that the number-theoretic functions which can (in principle) be computed by any deterministic, physical device are exactly those which can be computed by a Turing machine, and the correspon...

By the theory TT is meant the higher order predicate logic with the following recursively defined types: (1) 1 is the type ofindividuals and [] is the type ofthe truth values; (2) [# 1 ,..., n ] is the type ofthe predicates with arguments ofthe types #1 ,...,# n . The theory ITT described in this paper is an intensional version ofTT. The types ofITT are the same as the types ofTT, but the membership ofthe type 1 ofindividuals in ITT is an extension ofthe membership in TT. The extension consists ofallowing any higher order term, in which only variables oftype 1 have a free occurrence, to be a term oftype 1. This feature ofITT is motivated by a nominalist interpretation ofhigher order predication. In ITT both well-founded and non-well-founded recursive predicates can be defined as abstraction terms from which all the properties of the predicates can be derived without the use ofnon-logical axioms. The elementary syntax, semantics, and prooftheory for ITT are defined. A semantic consistency prooffor ITT is provided and the completeness proofofTakahashi and Prawitz for a version of TT without cut is adapted for ITT; a consequence is the redundancy of cut. 1.

machines and implementations The first definition of an abstract machine was given by Turing, in the classic [20]. Without repeating here the well-known definition (e.g., see [6]), we recall that each Turing machine M is equipped with a "semi-infinite tape" which it uses both to compute and also to communicate with its environment: To determine the value f(n) (if any) of the partial function f : N * N computed by M , we put n on the tape in some standard way, e.g., by placing n + 1 consecutive 1s at its beginning; we start the machine in some specified, initial, internal state q 0 and looking at the leftmost end of the tape; and we wait until the machine stops (if it does), at which time the value f(n) can be read off the tape, by counting the successive 1s at the left end. Turing argued that the number-theoretic functions which can (in principle) be computed by any deterministic, physical device are exactly those which can be computed by a Turing machine, and the corresponding version of this claim for partial functions has come to be known as the ChurchTuring Thesis, because an equivalent claim was made by Church at about the same time. Turing's brilliant analysis of "mechanical computation" in [20] and a huge body of work in the last sixty years has established the truth of the Church-Turing Thesis beyond reasonable doubt; it is of immense importance in the derivation of foundationally significant undecidability results from technical theorems about Turing machines, and it has been called "the first natural law of pure mathematics." Turing machines capture the notion of mechanical computability of numbertheoretic functions, by the Church-Turing Thesis, but they do not model faith- It has also been suggested that we do not need algorithms, only the equival...