Logic

A Problem Course in Mathematical Logic by Stefan Bilaniuk

By Stefan Bilaniuk

This is often the quantity II of a textual content for a problem-oriented undergraduate path in mathematical common sense. 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 path on those issues. quantity I, Propositional and First-Order common sense, covers the fundamentals of those themes throughout the Soundness, Completeness, and Compactness Theorems. details on availability and the stipulations below which this booklet can be used and reproduced are given within the preface.

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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). . and use it! 9. Suppose g is a k-place function and h is a k + 2-place function, both representable in Th(A). Then the k + 1place function f defined by primitive recursion from g and h is also representable in Th(A). 10.

PRIMITIVE RECURSIVE FUNCTIONS It isn’t too hard to show that A, and hence also α, are Turing computable. However, though it takes considerable effort to prove it, α grows faster with n than any primitive recursive function. (Try working out the first few values of α . . 12. 5 are Turing computable. If you are very ambitious, you can try to prove the following theorem. 13. 5 and f is any primitive recursive function. Then there is an n ∈ N such that for all k > n, α(k) > f(k). 14. 5 is not primitive recursive.

3. Every recursive function is Turing computable. We shall show that every Turing computable function is recursive later on. Similarly to primitive recursive relations we have the following. 4. A k-place relation P is said to be recursive (Turing computable) if its characteristic function χP is recursive (Turing computable). Since every recursive function is Turing computable, and vice versa, “recursive” is just a synonym of “Turing computable”, for functions and relations alike. 11 we have the following.

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