The Automation Of Proof By Mathematical Induction

Chapter appears in Handbook of Automated Reasoning Edited by: Alan Robinson and Andrei Voronkov ISBN: 978-0-444-50813-3This paper is a chapter of the Handbook of Automated Reasoning edited by Voronkov and Robinson. It describes techniques for automated reasoning in theories containing rules of mathematical induction. Firstly, inductive reasoning is defined and its importance fore reasoning about any form of repitition is stressed. Then the special search problems that arise in inductive theories are explained followed by descriptions of the heuristic methods that have been devised to solve these problems

Aspects of the constructive omega rule within automated deduction

In general, cut elimination holds for arithmetical systems with the w -rule, but not for systems with ordinary induction. Hence in the latter, there is the problem of generalisation, since arbitrary formulae can be cut in. This makes automatic theorem -proving very difficult. An important technique for investigating derivability in formal systems of arithmetic has been to embed such systems into semi- formal systems with the w -rule. This thesis describes the implementation of such a system. Moreover, an important application is presented in the form of a new method of generalisation by means of "guiding proofs" in the stronger system, which sometimes succeeds in producing proofs in the original system when other methods fail

What\u27s So Special About Kruskal\u27s Theorem and the Ordinal \u3cem\u3eT\u3c/em\u3e\u3csub\u3eo\u3c/sub\u3e? A Survey of Some Results in Proof Theory

This paper consists primarily of a survey of results of Harvey Friedman about some proof theoretic aspects of various forms of Krusal\u27s tree theorem, and in particular the connection with the ordinal Ƭo. We also include a fairly extensive treatment of normal functions on the countable ordinals, and we give a glimpse of Veblen Hierarchies, some subsystems of second-order logic, slow-growing and fast-growing hierarchies including Girard\u27s result, and Goodstein sequences. The central theme of this paper is a powerful theorem due to Kruskal, the tree theorem , as well as a finite miniaturization of Kruskal\u27s theorem due to Harvey Friedman. These versions of Kruskal\u27s theorem are remarkable from a proof-theoretic point of view because they are not provable in relatively strong logical systems. They are examples of so-called natural independence phenomena , which are considered by more logicians as more natural than the mathematical incompleteness results first discovered by Gödel. Kruskal\u27s tree theorem also plays a fundamental role in computer science, because it is one of the main tools for showing that certain orderings on trees are well founded. These orderings play a crucial role in proving the termination of systems of rewrite rules and the correctness of Knuth-Bandix completion procedures. There is also a close connection between a certain infinite countable ordinal called Ƭoand Kruskal\u27s theorem. Previous definitions of the function involved in this connection are known to be incorrect, in that, the function is not monotonic. We offer a repaired definition of this function, and explore briefly the consequences of its existence

Wittgenstein on the Foundations of Mathematics

In Part I, an attempt is made to survey the original source material on which any detailed assessment of Wittgenstein's remarks on the foundations of mathematics from his middle and later periods ought to be based. This survey is presented within the context of a sketch of Wittgenstein's biography, which also mentions some of the major developments in his thinking. In addition, certain main themes are emphasized; these have to do primarily with the Kantian aspects of Wittgenstein's thought and with his mysticism or the 'religious point of view'. In Part II, Kreisel's critique of Wittgenstein's remarks on the foundations of mathematics, which has been developed since 1958 in a series of published articles, receives close examination, and, in connection with this, different approaches to the philosophical investigation of mathematics are considered which represent genuine alternatives to Wittgenstein's approach. There are separate sections on Lakatos's Proofs and Refutations and Bourbaki's 'L'Architecture des Mathématiques'. Finally, besides a bibliography which surveys the reception of Wittgenstein's views on the foundations of mathematics, there are two substantial appendices, which are supplemental to Part I. The first of these gives the manuscript sources for typescripts 221 and 222-4, and the correspondences in both directions between these typescripts. The second appendix is part of a chronological version of von Wright's catalogue of Wittgenstein's papers, beginning in 1929

Finitism--an essay on Hilbert's programme

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 1991.Includes bibliographical references (p. 213-219).by David Watson Galloway.Ph.D