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09-decidability-2 - Another Proof We just saw a proof that...

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Another Proof We just saw a proof that languages that are not Turing-recognizable exist based on the fact that a decider for A T M cannot exist Let’s look at another proof that shows that some languages are not algorithmic. The proof proceeds by showing that the set of all Turing machines is countably infinite whereas the set of all languages is uncountable. Therefore, there exist some languages that are not recognized by a Turing machine. NOTE: Let 0 be the cardinality of the natural numbers and C the cardinality of the reals, then Cantor’s continuum hypothesis states that 0 < C . That is, the natural numbers are countably infinite whereas the reals are uncountable . a “There are fewer natural numbers than there are reals.” a The book has the classical proof of the uncountability of the reals based on diagonalization. – p. 1/
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Countable Sets Here are some simple examples of countable sets: n f ( n ) 1 1 2 2 . . . . . . k k n f ( n ) 1 10 2 20 . . . . . . k k 10 n f ( n ) 1 2 2 4 . . . . . . k k 2 n f ( n ) 1 1 2 3 . . . . . . k k 2 1 Observation: In all cases the mapping f between n and f ( n ) is one-to-one and onto, that is, it is bijective: Each value of n uniquely identifies a value of f ( n ) and there is no way to construct a member of the codomain of f that does not already appear in the correspondence.
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