Models for short sequences of measures in the cofinality-$ω$ constructible model
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arXiv
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866914685215834112 |
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| author | Ya'ar, Ur |
| author_facet | Ya'ar, Ur |
| contents | We investigate the relation between $C^{*}$, the model of sets constructible using first order logic augmented with the "cofinality-$ω$" quantifier, and "short" sequences of measures - sequences of measures of order $1$, which are shorter than their minimum. We show that certain core models for short sequences of measures are contained in $C^{*}$; we compute $C^{*}$ in a model of the form $L\left[\mathcal{U}\right]$ where $\mathcal{U}$ is a short sequence of measures, and in models of the form $L\left[\mathcal{U}\right]\left[G\right]$ where $G$ is generic for adding Prikry sequences to some of the measurables of $\mathcal{U}$; and prove that if there is an inner model with a short sequence of measures of order type $χ$, then there is such an inner model in $C^{*}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_04523 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Models for short sequences of measures in the cofinality-$ω$ constructible model Ya'ar, Ur Logic 03E45 (Primary) 03E70, 03E47, 03E55 (Secondary) We investigate the relation between $C^{*}$, the model of sets constructible using first order logic augmented with the "cofinality-$ω$" quantifier, and "short" sequences of measures - sequences of measures of order $1$, which are shorter than their minimum. We show that certain core models for short sequences of measures are contained in $C^{*}$; we compute $C^{*}$ in a model of the form $L\left[\mathcal{U}\right]$ where $\mathcal{U}$ is a short sequence of measures, and in models of the form $L\left[\mathcal{U}\right]\left[G\right]$ where $G$ is generic for adding Prikry sequences to some of the measurables of $\mathcal{U}$; and prove that if there is an inner model with a short sequence of measures of order type $χ$, then there is such an inner model in $C^{*}$. |
| title | Models for short sequences of measures in the cofinality-$ω$ constructible model |
| topic | Logic 03E45 (Primary) 03E70, 03E47, 03E55 (Secondary) |
| url | https://arxiv.org/abs/2109.04523 |