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223Gödel's incompleteness theoremsOxford University Press. 1992.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 fascinat…Read more
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117First-order logicSpringer Verlag. 1968.This completely self-contained study, widely considered the best book in the field, is intended to serve both as an introduction to quantification theory and as ...
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100Some unifying fixed point principlesStudia Logica 50 (1). 1991.This article is written for both the general mathematican and the specialist in mathematical logic. No prior knowledge of metamathematics, 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 …Read more
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84Uniform self-referenceStudia Logica 44 (4). 1985.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.
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71Some new double induction and superinduction principlesStudia Logica 49 (1). 1990.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.
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59Recursion theory for metamathematicsOxford University Press. 1993.This work is a sequel to the author's Godel's Incompleteness Theorems, though it can be read independently by anyone familiar with Godel's incompleteness theorem for Peano arithmetic. The book deals mainly with those aspects of recursion theory that have applications to the metamathematics of incompleteness, undecidability, and related topics. It is both an introduction to the theory and a presentation of new results in the field.
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55Diagonalization and self-referenceClarendon Press. 1994.This book presents a systematic, unified treatment of fixed points as they occur in Godels incompleteness proofs, recursion theory, combinatory logic, semantics, and metamathematics. Packed with instructive problems and solutions, the book offers an excellent introduction to the subject and highlights recent research.
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43Uniform Gentzen systemsJournal of Symbolic Logic 33 (4): 549-559. 1968.Generally speaking, it appears correct to say that in a formulation of first order logic in which a large number of connectives are taken as primitive which allows us to have our cake and eat it too.
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43Undecidability and recursive inseparabilityZeitschrift fur mathematische Logik und Grundlagen der Mathematik 4 (7-11): 143-147. 1958.
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36The tao is silentHarperSanFrancisco. 1977.The Tao Is Silent Is Raymond Smullyan's beguiling and whimsical guide to the meaning and value of eastern philosophy to westerners. "To me," Writes Smullyan, "Taoism means a state of inner serenity combined with an intense aesthetic awareness. Neither alone is adequate; a purely passive serenity is kind of dull, and an anxiety-ridden awareness is not very appealing." This is more than a book on Chinese philosophy. It is a series of ideas inspired by Taoism that treats a wide variety of subjects …Read more
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34Theories with Effectively Inseparable NucleiMathematical Logic Quarterly 6 (15-22): 219-224. 1960.
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34Set theory and the continuum problemClarendon Press. 1996.A lucid, elegant, and complete survey of set theory, this three-part treatment explores axiomatic set theory, the consistency of the continuum hypothesis, and forcing and independence results. 1996 edition.
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32Monadic Elementary Formal SystemsZeitschrift fur mathematische Logik und Grundlagen der Mathematik 7 (6): 81-83. 1961.
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31What is the Name of this Book?: The Riddle of Dracula and Other Logical PuzzlesPhilosophical Review 88 (3): 496. 1979.
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29Quine Willard Van Orman. Methods of logic. Revised edition, Henry Holt and Company, Inc., New York 1959, xx + 272 pp (review)Journal of Symbolic Logic 24 (3): 219-220. 1959.
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24Rudy Rucker. Mind tools. The five levels of mathematical reality. Houghton Mifflin Company, Boston1987, viii + 328 pp (review)Journal of Symbolic Logic 53 (4): 1254-1255. 1988.
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24Review: Willard Van Orman Quine, Methods of Logic (review)Journal of Symbolic Logic 24 (3): 219-220. 1959.
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20Forever undecided: a puzzle guide to GödelOxford University Press. 1987.Collects a variety of mathematics and logic puzzles, some based on the theorems of the mathematician Kurt Godel