We give limits defined in terms of abstract pointclasses of the amount of determinacy available in certain canonical inner models involving strong cardinals. We show for example:
Theorem A. Det(Π¹₁-IND) → there exists an inner model with a strong cardinal.
Theorem B. Det(AQI) → there exist type-1 mice and hence inner models with proper classes of strong cardinals.
where Π¹₁-IND(AQI) is the pointclass of boldface Π¹₁-inductive (respectively arithmetically quasi-inductive) sets of reals.
P. D. Welch. "Determinacy in strong cardinal models." J. Symbolic Logic 76 (2) 719 - 728, June 2011. https://doi.org/10.2178/jsl/1305810772