Determinacy in strong cardinal models. (English) Zbl 1221.03055

Welch, P. D.

J. Symb. Log. 76, No. 2, 719-728 (2011).

Summary: We give limits defined in terms of abstract point classes of the amount of determinacy available in certain canonical inner models involving strong cardinals. We show for example:
Theorem A. \(\text{Det}(\pmb\Pi ^1_{1}\text{-IND}) \Rightarrow \) there exists an inner model with a strong cardinal.
Theorem B. \(\text{Det}(\mathbf {AQI}) \Rightarrow \) there exist type-1 mice and hence inner models with proper classes of strong cardinals.
Here, \(\pmb\Pi ^1_{1}\text{-IND}\) (\(\mathbf {AQI}\)) is the point class of boldface \(\Pi ^1_{1}\)-inductive (respectively arithmetically quasi-inductive) sets of reals.

MSC:

03E60	Determinacy principles
03E15	Descriptive set theory
03E45	Inner models, including constructibility, ordinal definability, and core models

Keywords:

determinacy; inner models; strong cardinals; inductive sets of reals

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