Abstract
In this chapter we first introduce a standard cut-elimination procedure for the first-order logic and the \(\omega \)-logic, by which we see that the tree-height of the resulting cut-free proofs is bounded by a tower of exponential functions. Such a control of the tree-heights in cut-elimination is one of the most important results in ordinal analysis. A partial cut-elimination with a control holds as well for applied calculi. The partial result is sometimes helpful for us to analyze mathematical theories.
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Notes
- 1.
Axioms \(\varGamma ,\bar{L},L\) in \({\textit{\textbf{G}}}_{\omega }\) could be restricted to \(L\equiv (t\in X)\), cf. footnote 2. Therefore if \(\varGamma ,L\) is an axiom for a false closed literal L, then so is \(\varGamma \).
- 2.
Wlog we can assume \(\lnot C\) occurs in both uppersequents. Otherwise augment \(\lnot C\) by Weakening Lemma 3.1.
- 3.
For the finitary calculus \({\textit{\textbf{G}}}_{1}'\), \(\alpha <\omega \) suffices, but for an infinitary calculus such as \({\textit{\textbf{G}}}_{\omega }'\), we need the natural sum of ordinals.
- 4.
No renaming is needed here, but we need it to deal with a second-order \(\varepsilon \)-calculus in the Exercise 3.3.
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Arai, T. (2020). Cut-Elimination with Depths. 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_3
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