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Extra resources for Computability and Incompleteness

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E. p0 = 2). 28 CHAPTER 2. MODELS OF COMPUTATION We have seen that the set of primitive recursive functions is remarkably robust. But we will be able to do even more once we have developed an adequate means of handling sequences. I will identify finite sequences of natural numbers with natural numbers, in the following way: the sequence a0 , a1 , a2 , . . , ak corresponds to the number pa00 · pa11 · pa22 · . . · pkak +1 . I have added one to the last exponent, to guarantee that, for example, the sequences 2, 7, 3 and 2, 7, 3, 0, 0 have distinct numeric codes.

Following the handout, we will follow a few notational conventions: • When parentheses are left out, application takes place from left to right. For example, if M , N , P , and Q are terms, then M N P Q abbreviates (((M N )P )Q). • Again, when parentheses are left out, lambda abstraction is to be given the widest scope possible. From example, λx M N P is read λx (M N P ). • A period can be used to abstract multiple variables. For example, λxyz. M is short for λx λy λz M . For example, λxy. xxyxλz xz abbreviates λx λy ((((xx)y)x)λz (xz)).

You should think about what the definition means, and why the terminology is appropriate. The idea is that if S is the range of the computable function f , then S = {f (0), f (1), f (2), . }, and so f can be seen as “enumerating” the elements of S. e. the enumeration need not be in increasing order. In fact, f need not even be injective, so that the constant function f (x) = 0 enumerates the set {0}. Any computable set is computably enumerable. To see this, suppose S is computable. If S is empty, then by definition it is computably enumerable.

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