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.


03E45 Inner models, including constructibility, ordinal definability, and core models
03E55 Large cardinals
03E60 Determinacy principles
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