030094 Gödel: The Unprovability of the Consistency of Arithmetic (Kürbis, Skurt)

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Gödel’s first incompleteness theorem proved that if arithmetic is (omega) consistent, then it is not negation complete, that is, there is a sentence such that neither it nor its negation is provable in arithmetic. Gödel established this result by exhibiting a sentence of arithmetic, the so-called Gödel sentence, that is equivalent to the statement of its own unprovability in arithmetic. The second incompleteness theorem showed that if arithmetic is consistent, then it cannot prove the statement that expresses the consistency of arithmetic. This course is an introduction to all formal aspects of Gödel’s incompleteness theorems. We will begin with a recapitulation of fundamental results about first order logic, such as its completeness and the Löwenheim Skolem Theorem, and proceed to first order theories, in particular a fragment of number theory. Gödel’s method of the arithmetisation of syntax and its application to the formalisation of proofs in arithmetic will be presented in detail. We will then be ready to prove Gödel’s first incompleteness theorem. Afterwards we will consider the resources needed to prove the second incompleteness theorem. There will also be time to discuss the philosophical importance of Gödel’s results. | | | Literature: George Boolos: The Logic of Provability (Cambridge University Press 1993) Herbert B. Enderton: A Mathematical Introduction to Logic, 2nd edition (San Diego: Harcourt 2001) Eliot Mendelson: An Introduction to Mathematical Logic, 6th edition (Boca Raton: CRC Press 2015)