A Problem Course in Mathematical Logic by Stefan Bilaniuk

By Stefan Bilaniuk

This can be the amount II of a textual content for a problem-oriented undergraduate path 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 good judgment, covers the fundamentals of those themes during the Soundness, Completeness, and Compactness Theorems. info on availability and the stipulations less than which this e-book can be used and reproduced are given within the preface.

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Since recursive functions operate on integers, we first need to specify some way to code the tapes of Turing machines by integers. 3 in the process. As we did in those definitions, we shall stick to Turing machines with alphabet {1} for simplicity. 5. Suppose (i, s, a) is a tape position such that all but finitely many cells of a are blank. Let n be any positive integer such that ak = 0 for all k ∈ N with k > n. Then the code of (i, s, a) is n (i, s, a) = 2i 3s 5a0 7a1 11a2 . . pan+3 . 1. Consider the tape position (1, 2, 1001).

Nk ) = µm[g(n1, . . , nk , m) = 0]. Note. If there is no m such that g(n1 , . . , nk , m) = 0, then the unbounded minimalization of g is not defined on (n1 , . . , nk ). This is one reason we will occasionally need to deal with partial functions. If the unbounded minimalization of a computable function is to be computable, we have a problem even if we ask for some default output (0, say) to ensure that it is defined for all k-tuples. The obvious procedure which tests successive values of g to find the needed m will run forever if there is no such m, and the incomputability of the Halting Problem suggests that other procedure’s won’t necessarily succeed either.

Nk , m) = 0. This is often written as f(n1 , . . , nk ) = µm[g(n1, . . , nk , m) = 0]. Note. If there is no m such that g(n1 , . . , nk , m) = 0, then the unbounded minimalization of g is not defined on (n1 , . . , nk ). This is one reason we will occasionally need to deal with partial functions. If the unbounded minimalization of a computable function is to be computable, we have a problem even if we ask for some default output (0, say) to ensure that it is defined for all k-tuples. The obvious procedure which tests successive values of g to find the needed m will run forever if there is no such m, and the incomputability of the Halting Problem suggests that other procedure’s won’t necessarily succeed either.

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