2021/02/05 by Schlutzenberg, Farmer
#03E45 #03E55 #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2102.03359
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 T=<Tα>α<λ via Σ^*, consisting of normal trees Tα, each of length <Ω, there is a corresponding normal tree X via Σ with M^T_∞=MX_∞. Moreover, if there are no drops in model or degree along the main branches of these trees then the overall iteration maps i^T:M→ M^T_∞ and iX:M→ MX_∞ 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.