Increasing the second uniform indiscernible by strongly ssp forcing
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866929240960663552 |
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| author | De Bondt, Ben Velickovic, Boban |
| author_facet | De Bondt, Ben Velickovic, Boban |
| contents | We introduce a new and natural stationary set preserving forcing $\mathbb P^{c-c}(λ,μ)$ that (under $\mathsf{NS}_{ω_1}$ precipitous + existence of $H_θ^#$ for a sufficiently large regular $θ$) increases the second uniform indiscernible $\mathbf{u}_2$ beyond some given ordinal $λ$. The forcing $\mathbb P^{c-c}$ shares this property with forcings defined in [2] and [9]. As a main tool we use certain natural open two player games which are of independent interest, viz. the capturing games $\mathbf{G}_M^{cap}(X)$ and the catching-capturing games $\mathbf{G}_M^{c-c}(X)$. In particular, these games are used to isolate a special family of countable elementary submodels $M \prec H_θ$ that occur as side conditions in $\mathbb P^{c-c}$ and thus allow to control the forcing in a strong way. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_13797 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Increasing the second uniform indiscernible by strongly ssp forcing De Bondt, Ben Velickovic, Boban Logic Primary: 03E35, 03E55, 03E65, Secondary: 03E05 We introduce a new and natural stationary set preserving forcing $\mathbb P^{c-c}(λ,μ)$ that (under $\mathsf{NS}_{ω_1}$ precipitous + existence of $H_θ^#$ for a sufficiently large regular $θ$) increases the second uniform indiscernible $\mathbf{u}_2$ beyond some given ordinal $λ$. The forcing $\mathbb P^{c-c}$ shares this property with forcings defined in [2] and [9]. As a main tool we use certain natural open two player games which are of independent interest, viz. the capturing games $\mathbf{G}_M^{cap}(X)$ and the catching-capturing games $\mathbf{G}_M^{c-c}(X)$. In particular, these games are used to isolate a special family of countable elementary submodels $M \prec H_θ$ that occur as side conditions in $\mathbb P^{c-c}$ and thus allow to control the forcing in a strong way. |
| title | Increasing the second uniform indiscernible by strongly ssp forcing |
| topic | Logic Primary: 03E35, 03E55, 03E65, Secondary: 03E05 |
| url | https://arxiv.org/abs/2212.13797 |