So what is ? You wrote in your message of Sept 29:
The is compatible with all large cardinals. So is the
A second question. The version of you specified in your next message to me on Sept 29:
The (crude, uncut) is the statement that V is #-generated and if a sentence with absolute parameters holds in a cardinal-preserving, #-generated outer model then it holds in an inner model. It implies a strong failure of CH but is not known to be consistent.
does not even obviously imply . Perhaps you meant, the above together with ? If not then calling it is rather misleading. Either way it is closer to .
Anyway this explains my confusion, thanks.