By Stefan Bilaniuk

This can be the amount II of a textual content for a problem-oriented undergraduate direction in mathematical good judgment. It covers the fundamentals of computability, utilizing Turing machines and recursive features, and Goedel's Incompleteness Theorem, and will be used for a one semester direction on those subject matters. quantity I, Propositional and First-Order common sense, covers the fundamentals of those issues throughout the Soundness, Completeness, and Compactness Theorems. info on availability and the stipulations below which this publication can be used and reproduced are given within the preface.

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Prime(k) = pk , where p0 = 1 and pk is the kth prime if k ≥ 1. 4. Power(n, m) = k, where k ≥ 0 is maximal such that nk | m. 5. Length(n) = , where is maximal such that p | n. 6. Element(n, i) = ni , where n = pn1 1 . . pnk k (and ni = 0 if i > k). Using the representable functions and relations given above, we can represent the “history” function of any representable function . . 8. Suppose f is a k-place function representable in Th(A). ,nk ,0) F (n1, . . ,nk ,m) . . ,nk ,i) = pi i=0 is also representable in Th(A).

If f is a k-place partial function, the domain of f is the set dom(f) = { (n1 , . . , nk ) ∈ Nk | f(n1 , . . , nk ) is defined } . Similarly, the image of f is the set im(f) = { f(n1 , . . , nk ) | (n1 , . . , nk ) ∈ dom(f) } . In subsequent chapters we will also work with relations on the natural numbers. Recall that a k-place relation on N is formally a subset P of Nk ; P (n1 , . . , nk ) is true if (n1 , . . , nk ) ∈ P and false otherwise. In particular, a 1-place relation is really just a subset of N.

Nk , m). 2. For any constant a ∈ N, χ{a}(n) = 0 n=a 1 n = a. f(n1 , . . , nk ) (n1 , . . , nk ) = (c1 , . . , ck ) , if a (n1 , . . , nk ) = (c1 , . . , ck ) f is a primitive recursive k-place function and a, c1, . . , ck ∈ N are constants. 3. h(n1 , . . 8. Every primitive recursive function is Turing computable. Be warned, however, that there are computable functions which are not primitive recursive. We can extend the idea of “primitive recursive” to relations by using their characteristic functions.