Recursion Theory for Metamathematics
This work is a sequel to the author's Gödel's Incompleteness Theorems, though it can be read independently by anyone familiar with Gödel's incompleteness theorem for Peano arithmetic. The book deals mainly with those aspects of recursion theory that have applications to the metamathematics of incompleteness, undecidability, and related topics. It is both an introduction to the theory and a presentation of new results in the field.
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Recursive Enumerability and Recursivity
Undecidability and Recursive Inseparability
Generative Sets and Creative Systems
Double Generativity and Complete Effective Inseparability
Universal and Doubly Universal Systems
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a e w A1 and A2 argument arithmetic arithmetic set Assume hypothesis binary relations Chapter Corollary D.G. function disjoint pair double recursion theorem effective inseparability effectively a Rosser exact Rosser system exactly separates Exercise existential quantification first-order logic fixed point property formula H(v1 function for A1 Gödel number Gödel sentence Hence iteration theorem iterative function Kleene function Kleene pair Lemma metamathematical n-tuple natural numbers number h number n ordered pairs pair of r.e. Peano Arithmetic proof of Theorem Proposition provable formulas prove Theorem r.e. relation M(a r.e. sets recursive function f(z recursive function tſy recursive sets recursively inseparable refutable represents Rosser function semi-DSR sentential recursion property sets are representable Shepherdson strongly definable strongly separates superset Suppose system for binary system for sets Theorem 2.1 tion undecidable w-consistent weakly
Page viii - theorems of 1960 can be proved without appeal to the recursion theorem or any other fixed point argument.