Finitism, imperative programs and primitive recursion

Following the Crisis of Foundations Hilbert proposed to consider a finitistic form of arithmetic as mathematics’ safe core. This approach to finitism has often admitted primitive recursive function definitions as obviously finitistic, but some have advocated the inclusion of additional variants of recurrence, while others argued that, to the contrary, primitive recursion exceeds finitism. In a landmark essay, William Tait contested the finitistic nature of these extensions, due to their impredicativity, and advocated identifying finitism with primitive recursive arithmetic, a stance often referred to as Tait’s Thesis. However, a problem with Tait’s argument is that the recurrence schema has itself impredicative and non-finitistic facets, starting with an explicit reference to the functions being defined, which are after all infinite objects. It is therefore desirable to buttress Tait’s Thesis on grounds that avoid altogether any trace of concrete infinities or impredicativity. We propose here to do just that, building on the generic framework of [ 13]. We provide further evidence for Tait’s Thesis by outlining a proof of a purely finitistic version of Parsons’ theorem, whose intuitive gist is that finitistic reasoning is equivalent to finitistic computing.

Non-primitive Recursive Function Definitions

This paper presents an approach to the problem of introducing<br />non-primitive recursive function definitions in higher order logic. A<br />recursive specification is translated into a domain theory version, where<br />the recursive calls are treated as potentially non-terminating. Once we<br />have proved termination, the original specification can be derived easily.<br />A collection of algorithms are presented which hide the domain theory<br />from a user. Hence, the derivation of a domain theory specification<br />has been automated completely, and for well-founded recursive function<br />specifications the process of deriving the original specification from the<br />domain theory one has been automated as well, though a user must<br />supply a well-founded relation and prove certain termination properties<br />of the specification. There are constructions for building well-founded<br />relations easily.

Polynomially and superexponentially shorter proofs in fragments of arithmetic

In Parikh [71] it is shown that, if T is an r.e. consistent extension of Peano arithmetic PA, then, for each primitive recursive function g, there is a formula φ of PA such that(In the following, Proof T(z, φ) and Prov T(φ) denote the metalinguistic assertions that z codes a proof of φ in T and that φ is provable in T respectively, where ProofT(z, ┌φ┐) and ProvT(┌φ┐) are the formalizations of Proof T(z,φ) and ProvT(φ) respectively in the language of PA, ┌φ┐ denotes the Gödel number of φ and ┌φ┐ denotes the corresponding numeral. Also, for typographical reasons, subscripts will not be made boldface.) If g is a rapidly increasing function, we express (1) by saying that ProvT(┌φ┐) has a much shorter proof modulo g than φ. Parikh's result is based on the fact that a suitable formula A(x), roughly asserting that (1) holds with x in place of φ, has only provable fixed points. In de Jongh and Montagna [89], this situation is generalized and investigated in a modal context. There, a characterization is given of arithmetical formulas arising from modal formulas of a suitable modal language which have only provable fixed points, and Parikh's result is obtained as a particular case.

Pseudo-complements and ordinal logics based on consistency statements

Following [1] we write {n} for the nth recursively enumerable (re) set; that is, {n} = {x|VyT(n, x, y)}. By a “pair (T, α)” we mean a consistent re extension T of Peano arithmetic P and an RE-formula α which numerates the non-logical axioms of T in P [4]. Given a pair (T, α) and a particular formula which binumerates the Kleene T predicate in P, there can be defined a primitive recursive function Nα such that and which has the additional property that {Nα(Nα(n))} = ø for all n.

Incompleteness along paths in progressions of theories

We deal in the following with certain theories S, by which we mean sets of sentences closed under logical deduction. The basic logic is understood to be the classical one, but we place no restriction on the orders of the variables to be used. However, we do assume that we can at least express certain notions from classical first-order number theory within these theories. In particular, there should correspond to each primitive recursive function ξ a formula φ(χ), where ‘x’ is a variable ranging over natural numbers, such that for each numeral ñ, φ(ñ) expresses in the language of S that ξ(η) = 0. Such formulas, when obtained say by the Gödel method of eliminating primitive recursive definitions in favor of arithmetical definitions in +. ·. are called PR-formulas (cf. [1] §2 (C)).

Arithmetical representation of recursively enumerable sets

A set S of natural numbers is called recursively enumerable if there is a general recursive function F(x, y) such thatIn other words, S is the projection of a two-dimensional general recursive set. Actually, it is no restriction on S to assume that F(x, y) is primitive recursive. If S is not empty, it is the range of the primitive recursive functionwhere a is a fixed element of S. Using pairing functions, we see that any non-empty recursively enumerable set is also the range of a primitive recursive function of one variable.We use throughout the logical symbols ⋀ (and), ⋁ (or), → (if…then…), ↔ (if and only if), ∧ (for every), and ∨(there exists); negation does not occur explicitly. The variables range over the natural numbers, except as otherwise noted.Martin Davis has shown that every recursively enumerable set S of natural numbers can be represented in the formwhere P(y, b, w, x1 …, xλ) is a polynomial with integer coefficients. (Notice that this would not be correct if we replaced ≤ by <, since the right side of the equivalence would always be satisfied by b = 0.) Conversely, every set S represented by a formula of the above form is recursively enumerable. A basic unsolved problem is whether S can be defined using only existential quantifiers.

On axiomatizability within a system

Let C be the closure of a recursively enumerable set B under some relation R. Suppose there is a primitive recursive relation Q, such that Q is a symmetric subrelation of R (i.e. if Q(m, n), then Q(n, m) and R(m, n)), and such that, for each m ϵ B, Q(m, n) for infinitely many n. Then there exists a primitive recursive set A, such that C is the closure under R of A. For proof, note that , where f is a primitive recursive function which enumerates B, has the required properties. For each m ϵ B, there is an n ϵ A, such that Q(m, n) and hence Q(n, m); therefore the closure of A under Q, and hence that under R, includes B. Conversely, since Q is a subrelation of R, A is included in C. Finally, that A is primitive recursive follows from [2] p. 180.This observation can be applied to many formal systems S, by letting R correspond to the relation of deducibility in S, so that R(m, n) if and only if m is the Gödel number of a formula of S, or of a sequence of formulas, from which, together with axioms of S, a formula with the Gödel number n can be obtained by applications of rules of inference of S.