Abstract
This chapter introduces notion of definability in an arbitrary formalized theory T; more specifically, it explains under what condition a function on and to the natural numbers, or a set of natural numbers, is said to be definable in T. It shows that there is no consistent theory T in which a certain function D (the diagonal function) and a certain set V (the set of numbers correlated with valid sentences of T) are both definable. Hence, under some additional, though still very general assumptions, every consistent theory in which all recursive functions are definable is essentially undecidable. These results can serve as a general theoretical basis for the direct method in proofs of undecidability. The chapter also describes the formalized arithmetic N of natural numbers and some of its axiomatic subtheories—P, Q, and R. P is Peano's arithmetic; Q is a subtheory of P based upon a system of seven simple axioms; Ris a subtheory of Q with a very weak, though infinite, axiom system.
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