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Mathematical Association of America
Panu Raatikainen
John David Stone
I. Grattan-Guinness

Torkel Franzen

Godel's theorem : an incomplete guide to its use and abuse

Gödel's incompleteness theorem is one of the most well known mathematical results, but unfortunately this has led to it being mentioned in highly inappropriate ways. In Godel's theorem : an incomplete guide to its use and abuse Torkel Franzen examines various ways that this theorem has been wrongly quoted, and tries to set the reader straight on what it is really about. He looks at what has been said about incompleteness in physics, in theology and of course in various postmodern ramblings. There is also a chapter criticising attempts to use the incompleteness theorem in the philosophy of mind.

Franzen goes on to discuss the relationship of incompleteness with the infinite, as well as complexity and randomness. He also looks at Gödel's second incompleteness theorem and questions of consistency. Sometimes it seemed that he was being too pedantic, but on reflection I felt that this was justified. The important distinction is between a consistent and a sound theory. Notably, you get a consistent theory if you append to Peano Arithmetic (PA) an axiom expressing the inconsistency of PA.

I felt that this book would be most suited to those readers who have read the usual popular accounts of Gödel's incompleteness theorem and want to get its ramifications clearer in their minds, as well as those wanting a deeper view of the subject without venturing into too much technicality.