request for references

martdowd at martdowd at
Thu Nov 24 12:18:47 EST 2022

In a recent post Dymytro Taranovsky wrote
It is open whether 0^# is generic by tame forcing over some inner model M without 0^#.  However, because we have enough definability, such an M would have an iterated satisfaction relation for L (and hence cannot be L).
 I'm interested in this subject and would like to know if there are any references on it.
It's also of interest whether even an $L_\alpha$ with $\alpha$ countable, with an infinite set of indiscernibles, can be forced over $L$.  If this is not possible, it suggests (to me anyway) that the existence of such an $L_\alpha$ is false.  I have been considering looking in to whether there are non-standard models of ZFC where this is true.

Martin Dowd
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