A non-speedup result for the chain-antichain principle over a weak base theory
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908571361345536 |
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| author | Kowalik, Katarzyna W. |
| author_facet | Kowalik, Katarzyna W. |
| contents | We show that the theory $\mathsf{WKL}^*_0+\mathsf{CAC}$ is polynomially simulated by $\mathsf{RCA}_0^*$ with respect to $\forallΠ^0_3$ formulas. For the proof, we use the method of forcing interpretations and syntactically simulate a two-step model-theoretic argument, which involves construction of a restricted definable ultrapower, followed by a generic cut satisfying $\mathsf{CAC}$. Our result sharply contrasts with the previously known fact that $\mathsf{RCA}_0^*+\mathsf{RT}^2_2$ has non-elementary speedup over $\mathsf{RCA}_0^*$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_00323 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A non-speedup result for the chain-antichain principle over a weak base theory Kowalik, Katarzyna W. Logic 03F20, 03B30, 03F35, 05D10, 03F30, 03F25, 03H15 We show that the theory $\mathsf{WKL}^*_0+\mathsf{CAC}$ is polynomially simulated by $\mathsf{RCA}_0^*$ with respect to $\forallΠ^0_3$ formulas. For the proof, we use the method of forcing interpretations and syntactically simulate a two-step model-theoretic argument, which involves construction of a restricted definable ultrapower, followed by a generic cut satisfying $\mathsf{CAC}$. Our result sharply contrasts with the previously known fact that $\mathsf{RCA}_0^*+\mathsf{RT}^2_2$ has non-elementary speedup over $\mathsf{RCA}_0^*$. |
| title | A non-speedup result for the chain-antichain principle over a weak base theory |
| topic | Logic 03F20, 03B30, 03F35, 05D10, 03F30, 03F25, 03H15 |
| url | https://arxiv.org/abs/2510.00323 |