42+Slides--Godel%2C+Algorithms

42+Slides--Godel%2C+Algorithms - CS103 HO#42 Slides-Gdel,...

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CS103 HO#42 Slides--Gödel, Algorithms 5/9/11 1 In 1920, Hilbert proposed a research project that came to be know as Hilbert's Program . He wanted to achieve: --A formalization of all of mathematics --A proof that mathematics is consistent (no contradictions can be obtained). --A proof that mathematics is complete (all true statements can be proved in the formalism). --An algorithm for deciding the truth of falsity of any statement and thus the correctness of any proof. Gödel showed that this was not possible. Now, at last, The Incompleteness Theorem We say that a set of sentences T is formally complete if for any sentence S in the language, either S or ~S is provable from T. The set of sentences in question are the axioms of Peano arithmetic, and Gödel showed that Peano arithmetic is not formally complete. To begin, we need to realize that as well as writing expressions within a formal system, we can make statements about that system and those expressions. For example, the expression 2 + 3 = 5 belongs to mathematics, but the statement '2 + 3 = 5' is an arithmetical formula does not express an arithmetical fact and does not belong to the formal language of arithmetic. It belongs to meta-mathematics . The formulas x = x 0 = 0 belong to mathematics, but the statement 'x' is a variable belongs to meta-mathematics, since it characterizes a certain arithmetical sign as belonging to a specific class of signs, namely, variables. The following statement also belongs to meta-mathematics: The formula '0 = 0' is derivable from the formula 'x = x' by substituting the numeral '0' for the variable 'x'. Gödel devised a scheme that let him express meta-mathematical statements as purely arithmetical relations. This provided a way to make statements about the system inside the system itself. The starting point is a way to assign a unique number to every sign, formula (sequence of signs), and proof (sequence of formulas). We call these Gödel numbers . not 1 or 2 if. ..then. .. 3 there exists 4 equals 5 = zero 6 0 punctuation 10 , punctuation 9 ) punctuation 8 ( successor of 7 s Constant Gödel Usual sign number meaning
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CS103 HO#42 Slides--Gödel, Algorithms 5/9/11 2 0 11 x s0 13 y y 17 z Numerical Gödel Possible variable number substitution There are three kinds of variables. First are numerical variables like 'x', 'y', and 'z', for which numerals and numerical expressions can be substituted. Numerical variables are associated with primes greater than 10. 0 = 0 11 2 p ( x)(x = sy) 13 2 q p q 17 2 r Next are sentential variables like 'p', 'q', and 'r', for which formulas (sentences) can be substituted. Sentential variables are associated with squares of primes greater than 10. Numerical Gödel Possible variable number substitution Prime 11 3 P Composite 13 3 Q Greater than 17 3 R Next are predicate variables like 'P', 'Q', and 'R', for which predicates like 'Prime' or 'Greater than' can be substituted.
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42+Slides--Godel%2C+Algorithms - CS103 HO#42 Slides-Gdel,...

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