Abstract
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.
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- 1.
This is not actually necessary, since in ZFC one can talk directly about finite sequences and hence about \(\mathcal{L}\in \)-formulas (Remark 2 in 6.6), but we do so in order to maintain coherence with the exposition in 6.2.
- 2.
- 3.
An equivalent formalization of the prime factorization in PA using the β-function is \((\forall k\geq \underline{2})\exists u\exists n(k ={ \prod }_{i \leq n}{p}_{i}^{\langle \beta ui\rangle } \wedge \beta un\neq 0)\).
- 4.
- 5.
By such a T we mean that the proof steps of Solovay’s Theorem 5.2 not transgressing PA can be carried out in T. This does not yet imply the provability of the theorem itself. Which steps are transgressing PA is described in the following proof.
- 6.
In transitive models W the sentence in ‘ ’ (which with some encoding can be formulated in ℒ∈) is absolute, and therefore equivalent to the existence of a transitive model U ∈ W with U ⊨ α.
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Rautenberg, W. (2010). On the Theory of Self-Reference. In: A Concise Introduction to Mathematical Logic. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-1221-3_7
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