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Epsilon Numbers

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Ordinal Analysis with an Introduction to Proof Theory

Part of the book series: Logic in Asia: Studia Logica Library ((LIAA))

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Abstract

In this chapter let us begin with the celebrated result by G. Gentzen stating that the proof-theoretic ordinal of the first-order arithmetic PA is equal to the first epsilon number \(|\mathsf{PA}|=\varepsilon _{0}=\min \{\alpha >\omega : 2^{\alpha }=\alpha \}\). Our proof of the fact \(|\mathsf{PA}|\le \varepsilon _{0}\) is based on an elimination of first-order cut formulas. When we lower the cut degree by one, the tree-height of the resulting proof increases by one exponential. Thus the ordinal \(\alpha =|\mathsf{PA}|\) needs to be a fixed point \(2^{\alpha }=\alpha \) of an exponential function. Moreover \(|\mathsf{PA}|>\omega \) due to the presence of complete induction schema. The main result in this chapter is a slight generalization of Gentzen’s.

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Notes

  1. 1.

    The theory \(\text{ FiX }(\mathsf{PA})\) is also denoted by \(\widehat{ID}\) or \(\widehat{ID}(\mathsf{PA})\) in Sect. 5.4 below.

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  1. University of Tokyo, Tokyo, Japan

    Toshiyasu Arai

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Arai, T. (2020). Epsilon Numbers. In: Ordinal Analysis with an Introduction to Proof Theory. Logic in Asia: Studia Logica Library. Springer, Singapore. https://doi.org/10.1007/978-981-15-6459-8_4

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