Georg Kreisel

Informal Rigour and Completeness Proofs

Publisher Summary This chapter demonstrates the informal rigor and completeness proofs in mathematical philosophy. Informal rigor makes the analysis precise to eliminate the doubtful properties of the intuitive notions. The principal emphasis is on intuitive notions, which do not occur in ordinary mathematical practice but leads to new axioms for current notions. The difference between familiar independence results and the independence of the continuum hypothesis is discussed in the chapter; the difference is formulated in terms of higher order consequence. The chapter discusses the relation between intuitive logical consequence on the one hand and so-called “semantic syntactic” consequence on the other. Brouwers empirical propositions and proof considers a striking use of a new primitive to derive a purely mathematical assertion. The chapter explores the completeness questions for classical and intuitionist predicate logic. The problems of completeness involve informal rigor, at least when deciding completeness with respect to an intuitive notion of consequence.

On Weak Completeness of Intuitionistic Predicate Logic

Suppose the r i -placed relation symbols P i , 1 ≦ i ≦ k , are all the non-logical constants occurring in the closed formula , also written as , of Heytings predicate calculus (HPC). Then HPC is called complete for provided , i.e. Here D ranges over arbitrary species, and over arbitrary (possibly incompletely defined) subspecies of ;

A Notion of Mechanistic Theory

The notion in question is suggested by the words ‘mechanism’ or ‘machine’. Unlike the usual meaning of ‘mechanistic’, that is, deterministic in contrast to probabilistic, the notion here considered distinguishes among deterministic (and among probabilistic) theories.

A Survey of Proof Theory II

Abstract This paper explains recent work in proof theory from a neglected point of view. Proofs and their representations by formal derivations are treated as principal objects of study, not as mere tools for analyzing the consequence relation. Though the paper is principally expository it also contains some material not developed in the literature. In particular, adequacy conditions on criteria for the identity of proofs (in § 1c), and a reformulation of Godeľs second theorem in terms of the notion of canonical representation (in § 1d); the use of normalization, instead of normal form, theorems for a direct proof of closure under Churchs rule of the theory of species [in § 2a(ii)] and the useless-ness of bar recursive functionals for (functional) interpretations of systems containing Churchs thesis [in §2b(iii)]; the use of ordinal structures in a quantifier-free formulation of transfinite induction (in § 3); the irrelevance of axioms of choice to the explicit realizability of existential theorems both for classical and for Heytings logical rules (in § 4c) and some new uses of Heytings rules for analyzing the indefinite cumulative hierarchy of sets (in § 4d); a semantics for equational calculi suitable when terms are interpreted as rules for computation [in Appl. Ia(iii)], and, above all, an analysis of formalist semantics and its relation to realizability interpretations (in App. Ic). A less technical account of the present point of view is in [21].

A remark on free choice sequences and the topological completeness proofs

Below are collected some simple results on the theory of free choice sequences [4] or infinitely proceeding sequences (ips) as they are called in [5]. These results are not sufficient to settle the completeness problems for Heytings predicate calculus [4], as formulated in [12], neither in the strong nor in the weak sense. They are published because they lead to intuitionistically valid versions of completeness proofs which have appeared in the literature, particularly [16], [17], [18], and, with certain reservations, [1]. The problems considered below (except in §8) differ from those of [12] in the following respect: In [12] we were mainly concerned with formulae of Heytings predicate calculus which were not even classically provable, and showed that the calculus was complete with respect to certain classes of these formulae. The novel feature was that we established this completeness by means of intuitionistically valid methods , in fact methods which can be formalized in Heytings arithmetic. Here we are primarily concerned with formulae which are classically, but not intuitionistically, provable. As pointed out in [12], the nature of the completeness problem for such formulae is totally different: the class of predicates which provides the required counterexamples must come from a system (with an intuitionistically acceptable interpretation) which, unlike Heytings arithmetic, is not a subsystem of the corresponding classical system. An example of such a system is an extension FC, given below, of Heytings formalization of the theory of free choices.

Some Reasons for Generalizing Recursion Theory

Publisher Summary This chapter discusses some reasons for generalizing recursion theory (g.r.t.). The chapter corrects two common errors: the first is to suppose that there is just one use of g.r.t., the other is to suppose that there are so many (depending on unspecified purposes). Existing results on various generalizations are referred. The formulation and analysis of the aims of g.r.t. themselves are focussed. The metarecursion theory is discussed. The chapter presents the purposes of g.r.t. that include (a) advancing other parts of logic and mathematics; (b) understanding the mathematical character of ordinary recursion theory; and (c) analysis of a general concept of computation. Work on admissible sets constitutes a refinement because it concerns consequences of only special instances of the replacement scheme. An application of classical recursion theory is that this object is well-suited to the study of infinitary languages.

Functions, Ordinals, Species*

Publisher Summary This chapter discusses the proof theory of formal classical analysis, formulated as a two-sorted axiomatic theory with variables for natural numbers and number theoretic functions and the basic relations of equality and function evaluation. The analyses of the notion of free choice sequence in terms of which the continuous functions are defined have shown that the evident axioms for free choice sequences are unexpectedly weak ; in particular, for some formal theorems A, A’ cannot be derived from known axioms. The purpose of the present chapter is to summarize work in the past ten years on the proposal and to put the results in perspective by comparing them with other work on subsystems.

On the Kind of Data Needed for A Theory of Proofs

Part II of this article responds to the question implicit in the title: there are reasonably adequate data for a natural history of proofs, but not for a systematic science. The distinction between natural history and systematic or fundamental science is elaborated in the Introduction to Part II. Part I (§§1–3) prepares for Part II by listing some striking successes of early proof theory and the diminishing returns of later elaborations. The present article complements recent publications (Kreisel (1976) and (in press)) which stress negative aspects of current proof theory.

Mathematical logic: Tool and object lesson for science

The object lesson concerns the passage from the foundational aims for which various branches of modern logic were originally developed to the discovery of areas and problems for which logical methods are effective tools. The main point stressed here is that this passage did not consist of successive refinements, a gradual evolution by adaptation as it were, but required radical changes of direction, to be compared to evolution by migration. These conflicts are illustrated by reference to set theory, model theory, recursion theory, and proof theory. At the end there is a brief autobiographical note, including the touchy point to what extent the original aims of logical foundations are adequate for the broad question of the heroic tradition in the philosophy of mathematics concerned with the ‘nature’ of the latter or, in modern jargon, with the architecture of mathematics and our intuitive resonances to it.

Perspectives in the Philosophy of Pure Mathematics

Publisher Summary This chapter discusses perspectives in the philosophy of pure mathematics. As in all research philosophical analysis, of concepts and of aims, is needed in mathematics; particularly for the discovery of proper questions and conjectures, and for civilized formulations. In the chapter, the term, “Proofs” is understood to refer to abstract mental processes; for a complementary discussion referring to verbal arguments. Accordingly, the general topic of the symposium is here restricted to pure mathematics, where proofs are indeed central, while for many uses of mathematics in other sciences, the truth of a theorem is relevant, not the way it is proved.