超限栈的完全规范化
Full normalization for transfinite stacks
AI总结：
本文扩展正规迭代策略至超限栈，实现完全规范化，并保证迭代极限一致，结合Steel与作者的工作，并分析多迭代比较。
摘要：
我们描述了具有适当凝聚性质的正规迭代策略向正规树栈策略的扩展，并实现了完全规范化。给定一个正则的不可数基数$\Omega$和一个前小鼠$M$的$(m,\Omega+1)$-迭代策略$\Sigma$，使得$\Sigma$和$M$都具有适当的凝聚性质，我们将$\Sigma$扩展为最优-$(m,\Omega,\Omega+1)^*$-迭代博弈的策略$\Sigma^*$，使得对于所有$\lambda<\Omega$和所有通过$\Sigma^*$的栈$\vec{\mathcal{T}}=\left<\mathcal{T}_\alpha\right>_{\alpha<\lambda}$（由长度均为${<\Omega}$的正规树$\mathcal{T}_\alpha$组成），存在一个对应的通过$\Sigma$的正规树$\mathcal{X}$，满足$M^{\vec{\mathcal{T}}}_\infty=M^{\mathcal{X}}_\infty$。此外，如果这些树的主分支上没有模型或度数的下降，则整体迭代映射$i^{\vec{\mathcal{T}}}:M\to M^{\vec{\mathcal{T}}}_\infty$和$i^{\mathcal{X}}:M\to M^{\mathcal{X}}_\infty$一致。该构造是John Steel和作者工作的结合成果。我们还建立了$\Sigma^*$的一些进一步有用性质，并利用这些方法分析了通过共同策略对多个迭代进行比较的问题。
英文摘要：
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.
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