Before its discovery there was just one geometry and mathematics; the idea that another geometry existed was considered improbable. When Gauss discovered hyperbolic geometry, it is said that he did not publish anything about it out of fear of the "uproar of the Boeotians ", which would ruin his status as princeps mathematicorum Latin, "the Prince of Mathematicians". Main article: Begriffsschrift Begriffsschrift German for, roughly, "concept-script" is a book on logic by Gottlob Frege , published in , and the formal system set out in that book. Begriffsschrift is usually translated as concept writing or concept notation; the full title of the book identifies it as "a formula language , modeled on that of arithmetic , of pure thought.

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Before its discovery there was just one geometry and mathematics; the idea that another geometry existed was considered improbable. When Gauss discovered hyperbolic geometry, it is said that he did not publish anything about it out of fear of the "uproar of the Boeotians ", which would ruin his status as princeps mathematicorum Latin, "the Prince of Mathematicians".

Main article: Begriffsschrift Begriffsschrift German for, roughly, "concept-script" is a book on logic by Gottlob Frege , published in , and the formal system set out in that book. Begriffsschrift is usually translated as concept writing or concept notation; the full title of the book identifies it as "a formula language , modeled on that of arithmetic , of pure thought. Frege went on to employ his logical calculus in his research on the foundations of mathematics , carried out over the next quarter century.

Main article: Principia Mathematica Principia Mathematica, or "PM" as it is often abbreviated, was an attempt to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could in principle be proven.

As such, this ambitious project is of great importance in the history of mathematics and philosophy, [2] being one of the foremost products of the belief that such an undertaking may be achievable.

One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, which Russell discovered allowed for the construction of paradoxical sets. PM sought to avoid this problem by ruling out the unrestricted creation of arbitrary sets. The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an " effective procedure " e.

For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system.

The second incompleteness theorem, an extension of the first, shows that such a system cannot demonstrate its own consistency. Some authors refer to it as the "Equivalence Schema", a synonym introduced by Michael Dummett. T-theories form the basis of much fundamental work in philosophical logic , where they are applied in several important controversies in analytic philosophy.

By the completeness theorem of first-order logic , a statement is universally valid if and only if it can be deduced from the axioms, so the Entscheidungsproblem can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.

In , Alonzo Church and Alan Turing published independent papers [5] showing that a general solution to the Entscheidungsproblem is impossible, assuming that the intuitive notation of " effectively calculable " is captured by the functions computable by a Turing machine or equivalently, by those expressible in the lambda calculus.

This assumption is now known as the Church—Turing thesis.

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## Stephen Cole Kleene

In a note about writing the book, Kleene notes that up to , about 17, copies of the English version of his text were sold, as were thousands of various translations including a sold-out first print run of of the Russian translation. So this is a book with a quite pivotal influence on the education of later logicians, and on their understanding of the fundamentals of recursive function theory and the incompleteness theorems in particular. It is indeed still a pleasure to read or at least, it ought to be a pleasure for anyone interested enough in logic to be reading these pages. The Introduction to Metamathematics remains a really impressive achievement: and not one to be admired only from afar, either.

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## Metamathematics

He was awarded the Ph. In , he joined the mathematics department at the University of Wisconsin—Madison , where he spent nearly all of his career. After two years as an instructor, he was appointed assistant professor in While a visiting scholar at the Institute for Advanced Study in Princeton, —, he laid the foundation for recursion theory , an area that would be his lifelong research interest. In , he returned to Amherst College, where he spent one year as an associate professor of mathematics. He was an instructor of navigation at the U.

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## Introduction to Metamathematics

Samusar Summary verdict This book can still be warmly recommended as an enjoyable and illuminating presentation of fundamental material, written by someone who was himself so closely engaged in the early developments back in the glory days. This is all very attractively done. Request removal from index. Thomas Andrews k 11 Metamathematics and the Philosophy of Mind. History of Western Philosophy.

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