## Alan Turing: The EnigmaA gripping story of mathematics, science, computing, war history, cryptography, and homosexual persecution and liberation. Hodges tells how Turing's revolutionary idea of 1936-- the concept of a universal machine-- laid the foundation for the modern computer. Turing brought the idea to practical realization in 1945 with his electronic design. This work was directly related to Turing's leading role in breaking the German Enigma ciphers during World War II, a scientific triumph that was critical to Allied victory in the Atlantic. Despite his wartime service, Turing was eventually arrested, stripped of his security clearance, and forced to undergo a humiliating treatment program-- all for trying to live honestly in a society that defined homosexuality as a crime. This New York Times bestselling biography of the founder of computer science and artificial intelligence is the definitive account of an extraordinary mind and life. --Excerpted from 2014 version, published by Princeton University Press. |

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Page 83

There was a point of view that it was absurd to speak of the axioms of

Nothing could be more primitive than the integers. On the other hand, it could

certainly be asked whether there existed a kernel of fundamental properties of

the ...

There was a point of view that it was absurd to speak of the axioms of

**arithmetic**.Nothing could be more primitive than the integers. On the other hand, it could

certainly be asked whether there existed a kernel of fundamental properties of

the ...

Page 93

Another point was that the argument assumed that

fact,

more precisely, Gödel had shown that formalised

Another point was that the argument assumed that

**arithmetic**was consistent. If, infact,

**arithmetic**were inconsistent, then every assertion would be provable. Somore precisely, Gödel had shown that formalised

**arithmetic**must either be ...Page 143

In the same way, extending the axioms of

list of axioms, or by two, or by infinitely many infinite lists — there was again no

limit. The question was whether any of this would overcome the Gödel effect.

In the same way, extending the axioms of

**arithmetic**could be done by one infinitelist of axioms, or by two, or by infinitely many infinite lists — there was again no

limit. The question was whether any of this would overcome the Gödel effect.

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