Abstract
This chapter presents two topics that are of the greatest importance in any inquiry into the general nature of formalized theories, namely, the topics of incompleteness and undecidability. A theory T is incomplete if there is a sentence A in the language of T such that neither A nor ∼A is a theorem of T; T is undecidable if there is no effective procedure for determining in each case whether a formula A stated in the language of T is a theorem of T. A theory is essentially undecidable if it is consistent and all consistent extensions of T (including T itself) are undecidable. General Undecidability Theorem states that if T is a consistent theory in which all recursive functions and relations are representable, then T is essentially undecidable. Any consistent theory in which all recursive functions and relations are representable is undecidable. The chapter presents a number of general theorems concerning decidability, undecidability, and essential undecidability.
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