January 2011
Intermediate to advanced
192 pages
5h 34m
English
Content preview from Computability Theory
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120 Computability Theory
That is, our witness (our true unprovable sentence) asserts, in a sense, that it is itself
not in the axiomatic theory that yields T. It is saying (under this rather free translation),
“I am unprovable in this axiomatic theory.”
Digression: In 1931, G
¨
odel did not have the development of computability theory
available to him. Instead, he proceeded directly to an ingenious construction of a sen-
tence that could be freely translated as saying “I am unprovable in this axiomatic
theory.” This sentence had to be true (if it were false, we would have a provable false-
hood), and hence unprovable in the axiomatic theory. Even better ...
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