Truth and Provability Smullyan, Raymond M.
The Mathematical intelligencer,
03/2013, Letnik:
35, Številka:
1
Journal Article
Recenzirano
As is well known, in the first quarter of the twentieth century, there were two mathematical systems, Principia Mathematica, and Zermelo-Fraenkel Set Theory, which were so powerful that it was ...generally assumed that all mathematical questions could he decided--that is, either proved or disproved--in each of the systems. However, in 1933, the logician Kurt Godel startled the mathematical world by proving that this was not the case--that in each of these systems, as well as in a variety of related systems, there must be sentences that, though true, could not be proved within the systems. This celebrated result is known as Godel's Incompleteness Theorem. Closely related to this result is a theorem of the logician Alfred Tarski, roughly to the effect that in these systems truth of sentences of the system is not definable in the system. Tarski's theorem provides a proof of Godel's theorem, which in many ways is simpler than Gadels original proof. Here, Smullyan provides the essential ideas behind the proofs of the Godel and Tarski theorems.
Is there really a God, and if so, what is God actually like? Is there an afterlife, and if so, is there such a thing as eternal punishment for unrepentant sinners, as many orthodox Christians and ...Muslims believe? And is it really true that our unconscious minds are connected to a higher spiritual reality, and if so, could this higher spiritual reality be the very same thing that religionists call God? In his latest book, Raymond M. Smullyan invites the reader to explore some beautiful and some horrible ideas related to religious and mystical thought. In Part One, Smullyan uses the writings on religion by fellow polymath Martin Gardner as the starting point for some inspired ideas about religion and belief. Part Two focuses on the doctrine of Hell and its justification, with Smullyan presenting powerful arguments on both sides of the controversy. If God asked you to vote on the retention or abolition of Hell, he asks, how would you vote? Smullyan has posed this question to many believers and received some surprising answers. In the last part of his treasurable triptych, Smullyan takes up the beautiful and inspiring ideas of Richard Bucke and Edward Carpenter on Cosmic Consciousness. Readers will delight in Smullyan's observations on religion and in his clear-eyed presentation of many new and startling ideas about this most wonderful product of human consciousness.
Kurt Godel, the greatest logician of our time, startled the world of mathematics in 1931 with his Theorem of Undecidability, which showed that some statements in mathematics are inherently ...undecidable. His work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum theory brought him further worldwide fame. In this introductory volume, Raymond Smullyan, himself a well-known logician, guides the reader through the fascinating world of Godel's incompleteness theorems. The level of presentation is suitable for anyone with a basic acquaintance with mathematical logic. As a clear, concise introduction to a difficult but essential subject, the book will appeal to mathematicians, philosophers, and computer scientists.
This article is written for both the general mathematican and the specialist in mathematical logic. No prior knowledge of matemathematics, recursion theory or combinatory logic is presupposed, ...although this paper deals with quite general abstractions of standard results in those three areas. Our purpose is to show how some apparently diverse results in these areas can be derived from a common construction. In Section 1 we consider five classical fixed point arguments (or rather, generalizations of them) which we present as problems that the reader might enjoy trying to solve. Solutions are given at the end of the section. In Section 2 we show how all these solutions can be obtained as special cases of a single fixed point theorem. In Section 3 we consider another generalization of the five fixed point results of Section 1 and show that this is of the same strength as that of Section 2. In Section 4 we show some curious strengthenings of results of Section 3 which we believe to be of some interest on their own accounts.
Some new double analogues of induction and transfinite recursion are given which yields a relatively simple proof of a result of Robert Cowen, 2 which in turn is a strengthening of an earlier result ...of Smullyan 1, which in turn gives a unified approach to Zorn's Lemma, the transfinite recursion theorem and certain results about ordinal numbers.
Uniform Self-Reference Smullyan, Raymond M.
Studia logica,
12/1985, Letnik:
44, Številka:
4
Journal Article
Recenzirano
Self-referential sentences have played a key role in Tarski's proof 9 of the non-definibility of arithmetic truth within arithmetic and Gödel's proof 2 of the incompleteness of Peano Arithmetic. In ...this article we consider some new methods of achieving self-reference in a uniform manner.
FIXED POINTS AND SELF-REFERENCE Smullyan, Raymond M.
International Journal of Mathematics and Mathematical Sciences,
1984, Letnik:
1984, Številka:
2
Journal Article
Recenzirano
Odprti dostop
It is shown how Gödel's famous diagonal argument and a generalization of the recursion theorem are derivable from a common construation. The abstract fixed point theorem of this article is ...independent of both metamathematics and recursion theory and is perfectly comprehensible to the non-specialist.
PDF
