On the significance of parameters and the projective level in the Choice and Comprehension axioms
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909970012831744 |
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| author | Kanovei, Vladimir Lyubetsky, Vassily |
| author_facet | Kanovei, Vladimir Lyubetsky, Vassily |
| contents | We make use of generalized iterations of Jensen forcing to define a cardinal-preserving generic model of ZF for any $n\ge 1$ and each of the following four Choice hypotheses:
(1) $\text{DC}(\mathbfΠ^1_n)\land\neg\text{AC}_ω(\varPi^1_{n+1})\,;$
(2) $\text{AC}_ω(\text{OD})\land\text{DC}(\varPi^1_{n+1})\land \neg\text{AC}_ω(\mathbfΠ^1_{n+1});$
(3) $\text{AC}_ω\land\text{DC}(\mathbfΠ^1_n)\land\neg\text{DC}(\varPi^1_{n+1});$
(4) $\text{AC}_ω\land\text{DC}(\varPi^1_{n+1})\land\neg\text{DC}(\mathbfΠ^1_{n+1}).$
Thus if ZF is consistent and $n\ge1$ then each of these four conjunctions (1)--(4) is consistent with ZF.
As for the second main result, let PA$^0_2$ be the 2nd-order Peano arithmetic without the Comprehension schema $\text{CA}$. For any $n\ge1$, we define a cardinal-preserving generic model of ZF, and a set $M\subseteq\mathcal P(ω)$ in this model, such that $\langleω, M\rangle$ satisfies
(5) PA$^0_2$ + $\text{AC}_ω(\varSigma^1_{\infty})$ + $\text{CA}(\mathbfΣ^1_{n+1})$ + $\neg\text{CA}(\mathbfΣ^1_{n+1})$.
Thus $\text{CA}(\mathbfΣ^1_{n+1})$ does not imply $\text{CA}(\mathbfΣ^1_{n+2})$ in PA$^0_2$ even in the presence of the full parameter-free (countable) Choice $\text{AC}_ω(\varSigma^1_{\infty}).$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20098 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the significance of parameters and the projective level in the Choice and Comprehension axioms Kanovei, Vladimir Lyubetsky, Vassily Logic 03E25, 03E35, 03F35 (Primary), 03E15 (Secondary) We make use of generalized iterations of Jensen forcing to define a cardinal-preserving generic model of ZF for any $n\ge 1$ and each of the following four Choice hypotheses: (1) $\text{DC}(\mathbfΠ^1_n)\land\neg\text{AC}_ω(\varPi^1_{n+1})\,;$ (2) $\text{AC}_ω(\text{OD})\land\text{DC}(\varPi^1_{n+1})\land \neg\text{AC}_ω(\mathbfΠ^1_{n+1});$ (3) $\text{AC}_ω\land\text{DC}(\mathbfΠ^1_n)\land\neg\text{DC}(\varPi^1_{n+1});$ (4) $\text{AC}_ω\land\text{DC}(\varPi^1_{n+1})\land\neg\text{DC}(\mathbfΠ^1_{n+1}).$ Thus if ZF is consistent and $n\ge1$ then each of these four conjunctions (1)--(4) is consistent with ZF. As for the second main result, let PA$^0_2$ be the 2nd-order Peano arithmetic without the Comprehension schema $\text{CA}$. For any $n\ge1$, we define a cardinal-preserving generic model of ZF, and a set $M\subseteq\mathcal P(ω)$ in this model, such that $\langleω, M\rangle$ satisfies (5) PA$^0_2$ + $\text{AC}_ω(\varSigma^1_{\infty})$ + $\text{CA}(\mathbfΣ^1_{n+1})$ + $\neg\text{CA}(\mathbfΣ^1_{n+1})$. Thus $\text{CA}(\mathbfΣ^1_{n+1})$ does not imply $\text{CA}(\mathbfΣ^1_{n+2})$ in PA$^0_2$ even in the presence of the full parameter-free (countable) Choice $\text{AC}_ω(\varSigma^1_{\infty}).$ |
| title | On the significance of parameters and the projective level in the Choice and Comprehension axioms |
| topic | Logic 03E25, 03E35, 03F35 (Primary), 03E15 (Secondary) |
| url | https://arxiv.org/abs/2407.20098 |