On the Theory of Self-Reference

Part of the Universitext book series (UTX)


By self-reference we basically mean the possibility of talking inside a theory T about T itself or related theories. Here we can give merely a glimpse into this recently much advanced area of research; see e.g. [Bu]. We will prove Gödel’s second incompleteness theorem, Löb’s theorem, and many other results related to self-reference, while further results are discussed only briefly and elucidated by means of applications. All this is of great interest both for epistemology and the foundations of mathematics.

The mountain we first have to climb is the proof of the derivability conditions for \(\mathrm{PA}\) and related theories in 7.1, and the derivable Σ 1-completeness in 7.2. But anyone contented with leafing through these sections can begin straight away in 7.3; from then on we will just be reaping the fruits of our labor. However, one would forgo a real adventure in doing so, namely the fusion of logic and number theory in the analysis of \(\mathrm{PA}\). For a comprehensive understanding of self-reference, the material of 7.1 and 7.2 (partly prepared in Chapter 6) should be studied anyway.


Modal Logic Preference Order Recursive Function Derivability Condition Chinese Remainder Theorem 


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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Fachbereich Mathematik und InformatikBerlinGermany

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