57 Godel II - CS103A HO#57 Gdel II Gdel Numbering Gdel's...

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CS103A HO #57 Gödel II 3/12/08 1 Gödel's Incompleteness Theorem Kurt Gödel (1906 – 1978) ( x ) ( x = s y ) 8 4 11 9 8 11 5 7 13 9 2 · 3 · 5 · 7 · 11 · 13 · 17 · 19 · 23 · 29 This scheme allows us to represent every formula with a unique number. Given a number, we can determine whether it is a Gödel number, and if so, we can recover the formula, since every number has a unique prime factorization. Gödel Numbering ( x ) ( x = s y ) 8 4 11 9 8 11 5 7 13 9 2 · 3 · 5 · 7 · 11 · 13 · 17 · 19 · 23 · 29 S: Syntactic remark: S begins with '(' Arithmetic statement: The Gödel number is S is divisible by 2 8 but not by 2 9 Correspondence between syntactic properties of sentences and arithmetic properties of Gödel numbers Statement A: (p p) p Statement B: (p p) Syntactic remark: B is an initial part of A Arithmetic statement: The Gödel number of A is divisible by the Gödel number of B Correspondence between syntactic properties of sentences and arithmetic properties of Gödel numbers Gödel showed that all of the important syntactic notions of first-order logic can be represented in the language of Peano Arithmetic, such as: n is the Gödel number of a wff n is the Gödel number of a sentence n is the Gödel number of an axiom of Peano Arithmetic n is the Gödel number of a proof Correspondence between syntactic properties of sentences and arithmetic properties of Gödel numbers Now consider the statement 'The sequence of formulas with the Gödel number x is a proof of the formula with Gödel number z.' This statement is represented by a definite formula in the arithmetic calculus that expresses a purely arithmetic relation between x and z. So the truth or falsity of the statement hangs on whether x and z bear the proper arithmetic relationship.
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CS103A HO #57 Gödel II 3/12/08 2 Now consider the statement 'The sequence of formulas with the Gödel number x is a proof of the formula with Gödel number z.' This statement is represented by a definite formula in the arithmetic calculus that expresses a purely arithmetic relation between x and z. So the truth or falsity of the statement hangs on whether x and z bear the proper arithmetic relationship. We will denote this relationship with the name 'Dem', so we will write the sentence above as Dem(x,z) Note that even though the statement at the top is a meta-mathematical statement, there is a (complicated) formula for Dem in the arithmetic calculus. We need one more notation, concerning substitution for a variable: Suppose m is the Gödel number of the formula ( x)(x = sy) Subsitute in this formula for the variable with Gödel number 13 (i.e., y) the numeral for m. We get the formula ( x)(x = sm). We need one more notation, concerning substitution for a variable: Suppose m is the Gödel number of the formula ( x)(x = sy) Subsitute in this formula for the variable with Gödel number 13 (i.e., y) the numeral for m.
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