Approachable Free Subsets and Fine Structure Derived Scales

Shelah showed that the existence of free subsets over internally approachable subalgebras follows from the failure of the PCF conjecture on intervals of regular cardinals. We show that a stronger property called the Approachable Bounded Subset Property can be forced from the assumption of a cardinal $λ$ for which the set of Mitchell orders $\{ o(μ) \mid μ< λ\}$ is unbounded in $λ$. Furthermore, we study the related notion of continuous tree-like scales, and show that such scales must exist on all products in canonical inner models. We use this result, together with a covering-type argument, to show that the large cardinal hypothesis from the forcing part is optimal.

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