Bounds on the strength of ordinal definable determinacy in small admissible sets.(English)Zbl 1269.03052

The author presents upper and lower bounds on the large-cardinal strength of a statement, denoted by Det$$(*)$$, which asserts that a certain admissible set believes that every ordinal-definable class of reals is determined. Before defining Det$$(*)$$, we introduce some notation. For any real $$x\in{^\omega}\omega$$, let $$M_x=L_\alpha[x]$$, where $$\alpha$$ is the least ordinal so that $$L_\alpha[x]$$ is admissible and $$x\in L_\alpha[x]$$. Let $$\varphi(v)$$ be a first-order formula, in the language of set theory, in which ordinal parameters in $$M_x$$ are allowed to appear. Define the set of reals $$X_{\varphi}=\{y\in M_x\cap {^\omega}\omega : M_x\models \varphi(y)\}$$. Observe that $$X_\varphi$$ is a class in $$M_x$$. Now let $$M_x\models \text{Det(OD)}$$ represent the sentence: For every such formula $$\varphi$$ we have that $$M_x\models$$ “$$X_\varphi$$ is determined”. Finally, define Det$$(*)$$ to be the assertion $$(\exists x\in {^\omega}\omega) (M_x\models \text{Det(OD)})$$.
The author identifies a large-cardinal assumption LC$$(*)$$, which is just below that of “the existence of a premouse-like structure with a measurable cardinal $$\kappa$$ of Mitchell order $$\kappa^{++}$$ and $$\omega$$ successors”, and then proves that LC$$(*)$$ implies Det$$(*)$$. Thus, LC$$(*)$$ is an upper bound for Det$$(*)$$.
A. Lewis has previously shown that Det$$(*)$$ implies the existence of a countable model of ZFC with $$\lambda$$-many measurable cardinals (with $$\lambda$$ recursive). The author improves this lower bound by showing that Det$$(*)$$ implies the existence of a ZFC model with stronger sequences of measures.

MSC:

 3e+45 Inner models, including constructibility, ordinal definability, and core models 3e+55 Large cardinals 3e+60 Determinacy principles
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References:

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