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A robust proof-theoretic well-ordering
[article]
<span title="2022-02-28">2022</span>
<i >
arXiv
</i>
<span class="release-stage" >pre-print</span>
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<span title="2022-01-14">2022</span>
<span class="release-stage" >pre-print</span>
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2201.05284v1">arXiv:2201.05284v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/aucyscfdxfadblho3wmbbx4mnq">fatcat:aucyscfdxfadblho3wmbbx4mnq</a> </span>
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It is well-known that natural axiomatic theories are pre-well-ordered by logical strength, according to various characterizations of logical strength such as consistency strength and inclusion of Π^0_1 theorems. Though these notions of logical strength coincide for natural theories, they are not generally equivalent. We study analogues of these notions – such as Π^1_1-reflection strength and inclusion of Π^1_1 theorems – in the presence of an oracle for Σ^1_1 truths. In this context these
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... s coincide; moreover, we get genuine pre-well-orderings of axiomatic theories and may drop the non-mathematical quantification over "natural" theories.
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