Iterability for (transfinite) stacks
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arXiv
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| Format: | Preprint |
| Publié: |
2018
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| _version_ | 1866910907644248064 |
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| author | Schlutzenberg, Farmer |
| author_facet | Schlutzenberg, Farmer |
| contents | We establish natural criteria under which normally iterable premice are iterable for stacks of normal trees. Let $Ω$ be a regular uncountable cardinal. Let $m<ω$ and $M$ be an $m$-sound premouse and $Σ$ be an $(m,Ω+1)$-iteration strategy for $M$ (roughly, a normal $(Ω+1)$-strategy). We define a natural condensation property for iteration strategies, "inflation condensation". We show that if $Σ$ has inflation condensation then $M$ is $(m,Ω,Ω+1)^*$-iterable (roughly, $M$ is iterable for length $\leqΩ$ stacks of normal trees each of length ${<Ω}$), and moreover, we define a specific such strategy $Σ^{\mathrm{st}}$ and a reduction of stacks via $Σ^{\mathrm{st}}$ to normal trees via $Σ$. If $Σ$ has the Dodd-Jensen property and $\mathrm{card}(M)<Ω$ then $Σ$ has inflation condensation. We also apply some of the techniques developed to prove that if $Σ$ has strong hull condensation (introduced independently by John Steel) and $G$ is $V$-generic for an $Ω$-cc forcing, then $Σ$ extends to an $(m,Ω+1)$-strategy $Σ^+$ for $M$ with strong hull condensation, in the sense of $V[G]$. Moreover, this extension is unique. We deduce that if $G$ is $V$-generic for a ccc forcing then $V$ and $V[G]$ have the same $ω$-sound, $(ω,Ω+1)$-iterable premice which project to $ω$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1811_03880 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Iterability for (transfinite) stacks Schlutzenberg, Farmer Logic 03E45, 03E55 We establish natural criteria under which normally iterable premice are iterable for stacks of normal trees. Let $Ω$ be a regular uncountable cardinal. Let $m<ω$ and $M$ be an $m$-sound premouse and $Σ$ be an $(m,Ω+1)$-iteration strategy for $M$ (roughly, a normal $(Ω+1)$-strategy). We define a natural condensation property for iteration strategies, "inflation condensation". We show that if $Σ$ has inflation condensation then $M$ is $(m,Ω,Ω+1)^*$-iterable (roughly, $M$ is iterable for length $\leqΩ$ stacks of normal trees each of length ${<Ω}$), and moreover, we define a specific such strategy $Σ^{\mathrm{st}}$ and a reduction of stacks via $Σ^{\mathrm{st}}$ to normal trees via $Σ$. If $Σ$ has the Dodd-Jensen property and $\mathrm{card}(M)<Ω$ then $Σ$ has inflation condensation. We also apply some of the techniques developed to prove that if $Σ$ has strong hull condensation (introduced independently by John Steel) and $G$ is $V$-generic for an $Ω$-cc forcing, then $Σ$ extends to an $(m,Ω+1)$-strategy $Σ^+$ for $M$ with strong hull condensation, in the sense of $V[G]$. Moreover, this extension is unique. We deduce that if $G$ is $V$-generic for a ccc forcing then $V$ and $V[G]$ have the same $ω$-sound, $(ω,Ω+1)$-iterable premice which project to $ω$. |
| title | Iterability for (transfinite) stacks |
| topic | Logic 03E45, 03E55 |
| url | https://arxiv.org/abs/1811.03880 |