On principles between ∑1- and ∑2-induction, and monotone enumerations

Journal of Mathematical Logic 16 (1):1650004 (2016)

Abstract
We show that many principles of first-order arithmetic, previously only known to lie strictly between [Formula: see text]-induction and [Formula: see text]-induction, are equivalent to the well-foundedness of [Formula: see text]. Among these principles are the iteration of partial functions of Hájek and Paris, the bounded monotone enumerations principle by Chong, Slaman, and Yang, the relativized Paris–Harrington principle for pairs, and the totality of the relativized Ackermann–Péter function. With this we show that the well-foundedness of [Formula: see text] is a far more widespread than usually suspected. Further, we investigate the [Formula: see text]-iterated version of the bounded monotone iterations principle, and show that it is equivalent to the well-foundedness of the -height [Formula: see text]-tower [Formula: see text].
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DOI 10.1142/s0219061316500045
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References found in this work BETA

Ordinal Numbers and the Hilbert Basis Theorem.Stephen G. Simpson - 1988 - Journal of Symbolic Logic 53 (3):961-974.
On the Strength of Ramsey's Theorem Without S1-Induction.Keita Yokoyama - 2013 - Mathematical Logic Quarterly 59 (1-2):108-111.

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Reverse Mathematics and Colorings of Hypergraphs.Caleb Davis, Jeffry Hirst, Jake Pardo & Tim Ransom - 2019 - Archive for Mathematical Logic 58 (5-6):575-585.
Dickson’s Lemma and Weak Ramsey Theory.Yasuhiko Omata & Florian Pelupessy - 2019 - Archive for Mathematical Logic 58 (3-4):413-425.

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