Extensionality and choice in constructive mathematics

1982

In this dissertation I discuss Hilbert's thesis, the thesis that all acceptable mathematical arguments can be formalized using no logic stronger than first-order logic. In the first chapter, I present and criticize an argum~nt for Hilbert's thesis that is often found in the literature. The argument concludes that Hilbert's thesis is true since all mathematics is reducible to set theory and set theory is a first-order theory. I argue that the reduction mentioned is not enough to establish Hilbert's thesis unless we presuppose that Hilbert's thesis is true. In the second chapter I abstractly characterize logics and proof procedures. I then state Lindstrom's theorem (the theorem that, roughly, first-order logic is the only logic for which the completeness and Skolem-Lowenheim theorems are true) u5ing these characterizations of logics and proof procedur~s.

Russell: the Journal of Bertrand Russell Studies, 1993

T his book aims to make accessible to the general reader Russell's early work in logic and foundations of mathematics. The author holds, correctly I think, that understanding this work is essential for comprehension of Russell's more properly philosophical work. A conscientious attempt is thus made to explain mathematical and logical concepts without presupposing substantial background knowledge on the reader's part. The major portion of the book (four of its five chapters) consists of summaries, together with some relevant background and occasional critical discussion, of the following works: An Essay on the Foundations of Geometry, A Critical Exposition of the Philosophy ofLeibniz, "The Logic of Relations: with Some Applications to the Theory of Series", The Principles of Mathematics, "On Denoting", "Mathematical Logic as Based on the Theory of Types", Principia Mathematica, and the Introduction to the Second Edition of Principia Mathematica. Naturally, as the penultimate item assures, these summaries are often highly selective, although it was often unclear to me what the principles of selection were. With the exception of the Russell-Jourdain correspondence, no use is made of material Russell did not publish. (As indicated below, this leads to some serious shortcomings.) While there is fairly extensive, and sometimes uncritical, use of secondary literature, no use was apparently made of the 1980-81 issues of Synthese that were devoted to Russell's early work in philosophy and that contain important papers by Cocchiarella, Griffin and Hylton. Early on in the book, the author identifies two main strands of argument that "run through the whole of the book" (p. 3). The first of these is the contrast he aims to draw between Russell's plan to demonstrate the truth of mathematics and other approaches to the fo.undations of mathematics. It is surely correct to hold that, as early as The Principles ofMathematics (p. 4), Russell sees it as a virtue of logicism that it holds the claims of mathematics to be true. But this was not Russell's view during the years prior to about

2006

of accuracy. On the other hand, the constructivist claims that to know that a particular sequence of rational numbers converges is to know how to approximate the corresponding real number to an arbitrarily given degree of accuracy. Hence the constructivist rejects the law of excluded middle. Using the law of excluded middle one can also argue that every sequence of rational numbers that does not converge must diverge. However, one can not from the knowledge that a particular sequence of rational numbers does not converge construct the corresponding witness. The example just given illustrates the direct nature of constructive existence, as opposed to the indirect nature of classical existence, and shows why the law of excluded middle is not accepted as a law of constructive logic. Finally, I would like to thank my supervisor, Per Martin-Löf, for posing the problem of investigating how the double-negation interpretation operates on derivations and not only on formulas as well as for his continued guidance of my work. Without him, this thesis would never have come into existence. Jens Brage Reynolds (1972), and Plotkin (1975) for the foundations of CPS translations and Reynolds (1993) for the early history of continuations. To my knowledge there is no interpretation of classical logic in constructive logic that makes full use of the syntactic-semantic method of constructive type theory. With this thesis I hope to fill this gap. The subject of interpretations of classical logic in constructive logic began with the double negation interpretation of classical logic in minimal logic due to Kolmogorov (1925). The double negation interpretation was then followed by the interpretation of Peano arithmetic in Heyting arithmetic due to Gödel (1933) and the interpretation of classical logic in intuitionistic logic due to Kuroda (1951). Yet it was not until Griffin (1990) showed how to extend the formulae-as-types correspondence to classical logic that significant growth took place. His solution was to include operations on the flow of control, similar to call/cc of Scheme, into the notion of computation given by a simply typed call-by-value λ-calculus. After that Parigot (1992) introduced his λµ-calculus to realize classical proofs as programs. The λµ-calculus extended the simply typed λ-calculus with operators that can be used to model operations on the flow of control. The development then took the form of CPS translations of different λµ-calculi into different λ-calculi. See Ong (1996) and Ong and Stewart (1997) for call-by-value respectively call-byname CPS translations of Parigot's λµ-calculus into the simply typed λ-calculus. See Selinger (2001, p. 24) for an informal description of the semantics of the λµcalculus.

A principle, according to which any scientific theory can be mathematized, is investigated. That theory is presupposed to be a consistent text, which can be exhaustedly represented by a certain mathematical structure constructively. In thus used, the term " theory " includes all hypotheses as yet unconfirmed as already rejected. The investigation of the sketch of a possible proof of the principle demonstrates that it should be accepted rather a metamathematical axiom about the relation of mathematics and reality. Its investigation needs philosophical means. Husserl's phenomenology is what is used, and then the conception of " bracketing reality " is modelled to generalize Peano arithmetic in its relation to set theory in the foundation of mathematics. The obtained model is equivalent to the generalization of Peano arithmetic by means of replacing the axiom of induction with that of transfinite induction. Accepting or rejecting the principle, two kinds of mathematics appear differing from each other by its relation to reality. Accepting the principle, mathematics has to include reality within itself in a kind of Pythagoreanism. These two kinds are called in paper correspondingly Hilbert mathematics and Gödel mathematics. The sketch of the proof of the principle demonstrates that the generalization of Peano arithmetic as above can be interpreted as a model of Hilbert mathematics into Gödel mathematics therefore showing that the former is not less consistent than the latter, and the principle is an independent axiom. An information interpretation of Hilbert mathematics is involved. It is a kind of ontology of information. Thus the problem which of the two mathematics is more relevant to our being (rather than reality for reality is external only to Gödel mathematics) is discussed. An information interpretation of the Schrödinger equation is involved to illustrate the above problem.

We present a programme of research for pluralist formalisations — formalisations that involve proving results in more than one foundation. A foundation consists of two parts: a logical part that provides a notion of inference, and a non-logical part that provides the entities to be reasoned about. A logic-enriched type theory (LTT) is a formal system composed of such two separate parts. We show how LTTs may be used as the basis for a pluralist formalisation. We show how different foundations may be formalised as LTTs, and also describe a new method for proof reuse. If we know that a translation Φ exists between logic-enriched type theories (LTTs) S and T , and we have formalised a proof of a theorem α in S, we may wish to make use of the fact that Φ(α) is a theorem of T . We show how this is sometimes possible by writing a proof script MΦ . For any proof script Mα that proves a theorem α in S, if we change Mα so it first imports MΦ , then the resulting proof script will still parse, and will be a proof of Φ(α) in T . In this paper, we focus on the logical part of an LTT-framework and show how the above method of proof reuse is done for four cases of Φ: inclusion, the double negation translation, the A-translation, and the Russell-Prawitz modality. This work has been carried out using the proof assistant Plastic.