The proof-theoretic strength of Constructive Second-order set theories
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866908547416064000 |
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| author | Jeon, Hanul |
| author_facet | Jeon, Hanul |
| contents | In this paper, we define constructive analogues of second-order set theories, which we will call $\mathsf{IGB}$, $\mathsf{CGB}$, $\mathsf{IKM}$, and $\mathsf{CKM}$. Each of them can be viewed as $\mathsf{IZF}$- and $\mathsf{CZF}$-analogues of Gödel-Bernays set theory $\mathsf{GB}$ and Kelley-Morse set theory $\mathsf{KM}$. We also provide their proof-theoretic strengths in terms of classical theories, and we especially prove that $\mathsf{CKM}$ and full Second-Order Arithmetic have the same proof-theoretic strength. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_12854 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The proof-theoretic strength of Constructive Second-order set theories Jeon, Hanul Logic 03F25, 03E70, 03F50, 03F65 In this paper, we define constructive analogues of second-order set theories, which we will call $\mathsf{IGB}$, $\mathsf{CGB}$, $\mathsf{IKM}$, and $\mathsf{CKM}$. Each of them can be viewed as $\mathsf{IZF}$- and $\mathsf{CZF}$-analogues of Gödel-Bernays set theory $\mathsf{GB}$ and Kelley-Morse set theory $\mathsf{KM}$. We also provide their proof-theoretic strengths in terms of classical theories, and we especially prove that $\mathsf{CKM}$ and full Second-Order Arithmetic have the same proof-theoretic strength. |
| title | The proof-theoretic strength of Constructive Second-order set theories |
| topic | Logic 03F25, 03E70, 03F50, 03F65 |
| url | https://arxiv.org/abs/2312.12854 |