Incompleteness and Undecidability

  • Wolfgang RautenbergEmail author
Part of the Universitext book series (UTX)


Gödel’s fundamental results concerning the incompleteness of formal systems sufficiently rich in content, along with Tarski’s on the nondefinability of the notion of truth and Church’s on the undecidability of logic, as well as other undecidability results, are all based on essentially the same arguments. A widely known popularization of Gödel’s first incompleteness theorem runs as follows:

Consider a formalized axiomatic theory T that describes a given domain of objects \(\mathcal{A}\) in a manner that we hope is complete. Moreover, suppose that T is capable of talking in its language \(\mathcal{L}\) about its own syntax and proofs from its axioms. This is often possible if T has actually been devised to investigate other things (numbers or sets, say), namely by means of an internal encoding of the syntax of \(\mathcal{L}\). Then the sentence γ: “I am unprovable in T” belongs to \(\mathcal{L}\), where “I” refers precisely to the sentence γ (clearly, this possibility of self-reference has to be laid down in detail, which was the main work in [Gö2]). Then γ is true in \(\mathcal{A}\) but unprovable in T.


Recursive Function Axiom System Diophantine Equation Axiomatic Theory Incompleteness Theorem 
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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Fachbereich Mathematik und InformatikBerlinGermany

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