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Torkel Franzen, ?Godel's Theorem: An Incomplete Guide to Its Use and Abuse?

An excellent choice for courses that cover the philosophy of science and mathematics. Godel's Incompleteness Theorems were a revolution in mathematics and there were repercussions and misunderstandings that rippled out into other fields. The main theorem first appeared in an Austrian journal in 1931 and can be stated very simply.

In any consistent formal system S within which it is possible to perform a minimum amount of elementary arithmetic, there are statements that can neither be proved nor disproved.

The consequences are enormous, in that it means that in any system that can be used to perform arithmetic, there will be theorems that can never be verified as either true or false. In other words, some knowledge will forever be unattainable within that system. Of course, this does not preclude adding additional axioms that will allow other theorems to be proved.

Franzen does an excellent job in explaining the incompleteness theorems in a manner that can be understood by people with a limited knowledge of mathematics. While there are few places where a high school mathematics education is not sufficient to understand a more technical argument, it will be enough to understand and appreciate the theorems.

My favorite parts of the book were the sections devoted to "applications" of the incompleteness theorem outside of mathematics. Some examples are from religion, political science and philosophy. Godel's theorems are used to "prove" that no religion can contain a complete set of answers and that any constitution must of necessity be incomplete. Human thought is also interpreted in the context of the incompleteness theorems. The statement is:

Insofar as humans attempt to be logical, their thoughts form a formal system and are necessarily bound by Godel's theorem.

This statement and others related to the nature of human thought are examined in detail. The philosophy of Ayn Rand is also examined as a system that must of necessity be incomplete. This book would be an excellent supplemental text for a philosophy course where the nature of truth is examined. It would also be a very good choice for a course in the philosophy of mathematics.

-Published in Journal of Recreational Mathematics

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A more in-deep presentation is offered by Stephen Cole Kleene in its book "Introduction to Metamathematics".

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