Iterated club shooting and the stationary-logic constructible model
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866910045338337280 |
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| author | Ya'ar, Ur |
| author_facet | Ya'ar, Ur |
| contents | We investigate iterating the construction of $C(\mathtt{aa})$, the $L$-like inner model constructed using stationary-logic. We show that it is possible to force over generic extensions of $L$ to obtain a model of $V=C(\mathtt{aa})$, and to obtain models in which the sequence of iterated $C(\mathtt{aa})$s is decreasing of arbitrarily large order types. For this we prove distributivity and stationary-set preservation properties for countable iterations of club-shooting forcings using mutually stationary sets, and introduce the notion of mutually fat sets which yields better distributivity results even for uncountable iterations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_10247 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Iterated club shooting and the stationary-logic constructible model Ya'ar, Ur Logic 03E45 (Primary) 03E47, 03E55 (Secondary) We investigate iterating the construction of $C(\mathtt{aa})$, the $L$-like inner model constructed using stationary-logic. We show that it is possible to force over generic extensions of $L$ to obtain a model of $V=C(\mathtt{aa})$, and to obtain models in which the sequence of iterated $C(\mathtt{aa})$s is decreasing of arbitrarily large order types. For this we prove distributivity and stationary-set preservation properties for countable iterations of club-shooting forcings using mutually stationary sets, and introduce the notion of mutually fat sets which yields better distributivity results even for uncountable iterations. |
| title | Iterated club shooting and the stationary-logic constructible model |
| topic | Logic 03E45 (Primary) 03E47, 03E55 (Secondary) |
| url | https://arxiv.org/abs/2209.10247 |