Abstract
We fill an apparent gap in the literature by giving a short and self-contained proof that the ordinal of the theory \(\mathbf {RCA}_0 + \mathrm {WO}(\sigma )\) is \(\sigma ^\omega \), for any ordinal \(\sigma \) satisfying \(\omega \cdot \sigma = \sigma \) (e.g., \(\omega ^\omega \), \(\omega ^{\omega ^\omega }\), \(\varepsilon _0\)). Theories of the form \(\mathbf {RCA}_0 + \mathrm {WO}(\sigma )\) are of interest in Proof Theory and Reverse Mathematics because of their connections to a number of well-investigated combinatorial principles related to various subsystems of arithmetic.
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- 1.
For example, in proving that a \(\varPi ^1_1\)-version of Ramsey’s Theorem called the Adjacent Ramsey Theorem is equivalent to \(\mathrm {WO}(\varepsilon _0)\) over \(\mathbf {RCA}_0\), [4] Lemma 2.2 makes use of the false equivalence, over \(\mathbf {RCA}_0\), between \(\mathrm {WO}(\varepsilon _0)\) and the \(\varPi ^1_1\)-soundness of \(\mathbf {ACA}_0\). The presentation in the later [5] avoids this pitfall but establishes a slightly different result.
- 2.
They may also contain cuts with formulas \(R(t_1,\ldots ,t_k),\lnot R(t_1,\ldots ,t_k)\), where R is a symbol for a primitive recursive predicate. But these are entirely harmless.
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This publication was made possible through the support of a grant from the John Templeton Foundation (“A new dawn of intuitionism: mathematical and philosophical advances,” ID 60842). The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.
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Carlucci, L., Mainardi, L., Rathjen, M. (2019). A Note on the Ordinal Analysis of \(\mathbf {RCA}_0 + \mathrm {WO}(\mathbf {\sigma })\). In: Manea, F., Martin, B., Paulusma, D., Primiero, G. (eds) Computing with Foresight and Industry. CiE 2019. Lecture Notes in Computer Science(), vol 11558. Springer, Cham. https://doi.org/10.1007/978-3-030-22996-2_13
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