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Unformatted text preview: on p. 131, it implies that ℒ g is recursive. Problem 2. Prove that the subsets of distinct primes form an uncountable set. Proof. Denote the ± th prime as ² ³ . It is easy to show that there is a bijection between ℕ and primes ´ ∶ µ1,2,3,4,. ..,±,. .. ¶ U µ2,3,5,7,. .., ² ³ ,... ¶ . Therefore, primes are countable. Thus the problem is equivalent to asking whether a function · exists such that · ∶ ℕ U 2 ℕ is a bijection. Cantor’s theory says no such · exists. Hence the subsets of distinct primes do not form a countable set ....
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- Fall '11