An undecidable arithmetical statement

Abstract

Publisher Summary This chapter provides an alternative proof of the existence of formally undecidable sentences. Instead of the arithmetization of syntax and the diagonal process which were used by Godel, some simple set-theoretic lemmas and the Skolem-Lowenheim theorem are used. The undecidability of the sentence to be constructed is independent of whether the absolute notion of integers or the relative one is accepted. The proof of undecidability rests on the axioms of the Zermelo-Fraenkel set-theory including the axiom of choice and an additional axiom ensuring the existence of at least one inaccessible aleph. The chapter presents notations and terminology, and discusses set-theoretical and arithmetical formulae. A set-theoretical formula is an expression built up from elementary expressions with the help of the logical connectives and quantifiers. The chapter discusses the reduction of certain set-theoretical formulae to the arithmetical formulae. It further deals with the construction of an undecidable sentence.