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Aug 8, 2017 at 13:48 comment added Gro-Tsen (Sorry I had missed your edit. I don't know why MO didn't notify me.) Great answer, and thanks again for all the clarifications!
Aug 8, 2017 at 13:46 vote accept Gro-Tsen
Aug 7, 2017 at 14:33 history edited Philip Welch CC BY-SA 3.0
To address Gro-Tsen's queries (see comment below) and intended to complete both the original Question and its reformulation.
Aug 6, 2017 at 18:23 comment added Gro-Tsen Thanks, this clarifies a lot of things, and I now realize why my reasoning as to why $L_γ\mathrel{\preceq_1}L_β$ with $γ<β$ implies $γ$ not p.r.-recognizable, was incorrect. I do have three small questions which you might still clarify, though: • (A) Can you remind me (or provide a reference) why the smallest $α$ s.t. $L_α\models A$ is p.r.-recognizable? • (B) Could you detail your final comment? • (C) Do you think that the first p.r.-reflecting is what you called $β_0$ and that the first p.r.-recognizable after that is $β$ or do you expect complications?
Aug 6, 2017 at 17:32 history edited Philip Welch CC BY-SA 3.0
A negation (in English) was removed. A pointer to a lter part of the answer was added.
Aug 6, 2017 at 16:03 history answered Philip Welch CC BY-SA 3.0