Abstract
In this chapter we present an approach to the classical results due to Gentzen. In 1936, Gentzen proved consistency of arithmetic using transfinite induction up to \(\varepsilon _0\). We show how to prove transfinite induction up to any \(\alpha <\varepsilon _0\) within Peano Arithmetic. The proofs exhibit a tradeoff between a strength of a needed fragment of Peano arithmetic and the length of induction up to a given ordinal \(\alpha \), and the complexity of an induction formula. We introduce the method of idicators to show independence of combinatorial statements from fragments of Peano arithmetics.
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Notes
- 1.
Unfortunately, these parts of the book presenting the work from [9] were not finished.
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Kotlarski, H. (2019). Transfinite Induction. In: Adamowicz, Z., Bigorajska, T., Zdanowski, K. (eds) A Model–Theoretic Approach to Proof Theory. Trends in Logic, vol 51. Springer, Cham. https://doi.org/10.1007/978-3-030-28921-8_4
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DOI: https://doi.org/10.1007/978-3-030-28921-8_4
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