Full normalization for transfinite stacks
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866914716215934976 |
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| author | Schlutzenberg, Farmer |
| author_facet | Schlutzenberg, Farmer |
| contents | We describe the extension of normal iteration strategies with appropriate condensation properties to strategies for stacks of normal trees, with full normalization. Given a regular uncountable cardinal $Ω$ and an $(m,Ω+1)$-iteration strategy $Σ$ for a premouse $M$, such that $Σ$ and $M$ both have appropriate condensation properties, we extend $Σ$ to a strategy $Σ^*$ for the optimal-$(m,Ω,Ω+1)^*$-iteration game such that for all $λ<Ω$ and all stacks $\vec{\mathcal{T}}=\left<\mathcal{T}_α\right>_{α<λ}$ via $Σ^*$, consisting of normal trees $\mathcal{T}_α$, each of length ${<Ω}$, there is a corresponding normal tree $\mathcal{X}$ via $Σ$ with $M^{\vec{\mathcal{T}}}_\infty=M^{\mathcal{X}}_\infty$. Moreover, if there are no drops in model or degree along the main branches of these trees then the overall iteration maps $i^{\vec{\mathcal{T}}}:M\to M^{\vec{\mathcal{T}}}_\infty$ and $i^{\mathcal{X}}:M\to M^{\mathcal{X}}_\infty$ agree. The construction is the result of a combination of work of John Steel and of the author.
We also establish some further useful properties of $Σ^*$, and use the methods to analyze the comparison of multiple iterates via a common such strategy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_03359 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Full normalization for transfinite stacks Schlutzenberg, Farmer Logic 03E45, 03E55 We describe the extension of normal iteration strategies with appropriate condensation properties to strategies for stacks of normal trees, with full normalization. Given a regular uncountable cardinal $Ω$ and an $(m,Ω+1)$-iteration strategy $Σ$ for a premouse $M$, such that $Σ$ and $M$ both have appropriate condensation properties, we extend $Σ$ to a strategy $Σ^*$ for the optimal-$(m,Ω,Ω+1)^*$-iteration game such that for all $λ<Ω$ and all stacks $\vec{\mathcal{T}}=\left<\mathcal{T}_α\right>_{α<λ}$ via $Σ^*$, consisting of normal trees $\mathcal{T}_α$, each of length ${<Ω}$, there is a corresponding normal tree $\mathcal{X}$ via $Σ$ with $M^{\vec{\mathcal{T}}}_\infty=M^{\mathcal{X}}_\infty$. Moreover, if there are no drops in model or degree along the main branches of these trees then the overall iteration maps $i^{\vec{\mathcal{T}}}:M\to M^{\vec{\mathcal{T}}}_\infty$ and $i^{\mathcal{X}}:M\to M^{\mathcal{X}}_\infty$ agree. The construction is the result of a combination of work of John Steel and of the author. We also establish some further useful properties of $Σ^*$, and use the methods to analyze the comparison of multiple iterates via a common such strategy. |
| title | Full normalization for transfinite stacks |
| topic | Logic 03E45, 03E55 |
| url | https://arxiv.org/abs/2102.03359 |