Introduction to Metamathematics - Softcover

Softcover

ISBN 10: 0923891579 ISBN 13: 9780923891572

Verlag: Ishi Press, 2009

Inhaltsangabe

Stephen Cole Kleene was one of the greatest logicians of the twentieth century and this book is the influential textbook he wrote to teach the subject to the next generation. It was first published in 1952, some twenty years after the publication of Gödel's paper on the incompleteness of arithmetic, which marked, if not the beginning of modern logic, at least a turning point after which “nothing was ever the same.” Kleene was an important figure in logic, and lived a long full life of scholarship and teaching. The 1930s was a time of creativity and ferment in the subject, when the notion of “computable” moved from the realm of philosophical speculation to the realm of science. This was accomplished by the work of Kurt Göde1, Alan Turing, and Alonzo Church, who gave three apparently different precise definitions of “computable”. When they all turned out to be equivalent, there was a collective realization that this was indeed the “right notion”. Kleene played a key role in this process. One could say that he was “there at the beginning” of modern logic. He showed the equivalence of lambda calculus with Turing machines and with Gödel's recursion equations, and developed the modern machinery of partial recursive functions. This textbook played an invaluable part in educating the logicians of the present. It played an important role in their own logical education.

Über diesen Titel

Reseña del editor

Metamathematics is `mathematics used to study mathematics', or it involves the application of a philosophy of mathematics. The first part of this general description appears tautological, or is perhaps open to Bertrand Russell's and Alfred Whitehead's types of antimonies (e.g., "the of all sets is not a set"), as described in their famous "Principia Mathematica". An alternative, non-circular definition is as follows: Metamathematics is the study of metatheories of standard theories in mathematics, or about mathematical--not `purely logical'-- theories. Thus, in Encyclopædia Britannica, metatheory is defined as a " ,MT, the subject matter of which is another theory, T . A finding proved in the former (MT) that deals with the latter (T) is known as a metatheorem " (cited from Metatheory-Encyclopædia Britannica Online). Thus, a major part of metamathematics deals with: metatheorems, that is " about theorems", meta-propositions about propositions, metatheories about mathematical proofs (that of course utilize logic, but also are based upon fundamental mathematics concepts), and so on. Meta-mathematical metatheorems about mathematics itself were originally differentiated from ordinary mathematical theorems in the 19th century, to focus on what was then called the foundational crisis of mathematics. Richard's paradox concerning certain 'definitions' of real numbers in the English language is an example of the sort of contradictions which can easily occur if one fails to distinguish between mathematics and metamathematics. Bertrand Russell's and Alfred Whitehead's type of paradoxes is yet another important example of possible contradictions due to such failures in the 'old' set theory.

Details zur Ausgabe

• Verlag: Ishi Press
• Erscheinungsdatum: 2009
• Sprache: Englisch
• ISBN 10: 0923891579
• ISBN 13: 9780923891572
• Einband: Tapa blanda
• Anzahl der Seiten: 572
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: Nicht verfügbar