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 Sep 24 awarded Autobiographer Apr 5 awarded Yearling Aug 29 awarded Supporter Aug 29 comment About Sigma_1 definability Thank you for your comment! Aug 29 comment About Sigma_1 definability Thank you so much for your answer, it is very helpful for me. I have been on a travel without internet these days, so it is so late to reply your answer. Aug 29 accepted About Sigma_1 definability Aug 19 awarded Editor Aug 19 revised About Sigma_1 definability added 7 characters in body Aug 19 asked About Sigma_1 definability Aug 4 accepted Definability of the direct limit Aug 3 comment Definability of the direct limit I just read the remark on iterated HOD classes in your paper. For the iterated ultrapower case, the graph is definable, but this does not happen in iterated HOD case. Thanks for your helpful answer again. Aug 3 comment Definability of the direct limit Thanks for your kind answer. If the entire system $$and$$ are definable, then pick any representation of the direct limit, it is definable. But I still not well understand why the entire system is definable? $$is a sequence of classes, in general, an infinite sequence of classes need not be definable. But I have a feeling this sequence$$ is similar to a sequence defined by induction, so whether "a sequence of classes defined by induction" is indeed definable? Aug 3 asked Definability of the direct limit Jul 21 awarded Nice Question Apr 7 comment A least model contains a given model and a given set Thanks for your comments. I understand. If M is a proper class, then using closure of Godel's operations, there is a "closure" of $M\cup\{x\}$ as Jech' 13.24, which is an inner model of ZF. But if M is a set, then from the above link answer, such M[x] need not exist. Apr 4 awarded Scholar Apr 4 accepted A subgroup intersects every conjugacy class Apr 4 comment A subgroup intersects every conjugacy class The example of upper triangular matrix is so standard, thank you for your example! Apr 4 comment A subgroup intersects every conjugacy class Thank you for your answer. Apr 4 asked A least model contains a given model and a given set