The behavior of higher proof theory I: Case $Σ^1_2$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912886115270656 |
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| author | Jeon, Hanul |
| author_facet | Jeon, Hanul |
| contents | Walsh [MR4525964, Zbl 1569.03151] has shown that comparing proof-theoretic ordinals is equivalent to comparing $Π^1_1$-consequence comparison and $Π^1_1$-reflection comparison, all modulo true $Σ^1_1$-sentences. In this paper, we prove the analogous result for $Σ^1_2$-consequences modulo true $Π^1_2$-sentences, that is, the equivalence between $Σ^1_2$-proof-theoretic ordinal comparison, $Σ^1_2$-consequence comparison, and $Σ^1_2$-reflection comparison, all modulo true $Π^1_2$-sentences. We also examine the connection between $Σ^1_2$-proof-theoretic ordinal and $Σ^1_2$-analogue of the robust reflection rank in Pakhomov-Walsh [MR4362917, Zbl 1511.03018] |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_03801 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The behavior of higher proof theory I: Case $Σ^1_2$ Jeon, Hanul Logic 03F03, 03F15, 03F35, 03E10 Walsh [MR4525964, Zbl 1569.03151] has shown that comparing proof-theoretic ordinals is equivalent to comparing $Π^1_1$-consequence comparison and $Π^1_1$-reflection comparison, all modulo true $Σ^1_1$-sentences. In this paper, we prove the analogous result for $Σ^1_2$-consequences modulo true $Π^1_2$-sentences, that is, the equivalence between $Σ^1_2$-proof-theoretic ordinal comparison, $Σ^1_2$-consequence comparison, and $Σ^1_2$-reflection comparison, all modulo true $Π^1_2$-sentences. We also examine the connection between $Σ^1_2$-proof-theoretic ordinal and $Σ^1_2$-analogue of the robust reflection rank in Pakhomov-Walsh [MR4362917, Zbl 1511.03018] |
| title | The behavior of higher proof theory I: Case $Σ^1_2$ |
| topic | Logic 03F03, 03F15, 03F35, 03E10 |
| url | https://arxiv.org/abs/2406.03801 |