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| Main Author: | |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2206.08147 |
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| _version_ | 1866908783661285376 |
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| author | Goto, Tatsuya |
| author_facet | Goto, Tatsuya |
| contents | Goldstern showed in his 1993 paper that the union of a real-parametrized, monotone family of Lebesgue measure zero sets has also Lebesgue measure zero provided that the sets are uniformly $\boldsymbolΣ^1_1$. Our aim is to study to what extent we can drop the $\boldsymbolΣ^1_1$ assumption. We show Goldstern's principle for the pointclass $\boldsymbolΠ^1_1$ holds. We show that Goldstern's principle for the pointclass of all subsets is consistent with $\mathsf{ZFC}$ and show its negation follows from $\mathsf{CH}$. Also we prove that Goldstern's principle for the pointclass of all subsets holds both under $\mathsf{ZF} + \mathsf{AD}$ and in Solovay models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_08147 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Goldstern's principle about unions of null sets Goto, Tatsuya Logic Goldstern showed in his 1993 paper that the union of a real-parametrized, monotone family of Lebesgue measure zero sets has also Lebesgue measure zero provided that the sets are uniformly $\boldsymbolΣ^1_1$. Our aim is to study to what extent we can drop the $\boldsymbolΣ^1_1$ assumption. We show Goldstern's principle for the pointclass $\boldsymbolΠ^1_1$ holds. We show that Goldstern's principle for the pointclass of all subsets is consistent with $\mathsf{ZFC}$ and show its negation follows from $\mathsf{CH}$. Also we prove that Goldstern's principle for the pointclass of all subsets holds both under $\mathsf{ZF} + \mathsf{AD}$ and in Solovay models. |
| title | Goldstern's principle about unions of null sets |
| topic | Logic |
| url | https://arxiv.org/abs/2206.08147 |