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Sep 3, 2015 at 6:54 comment added Yair Hayut You can change the set of ordinals which are countable in HOD by forcing, and this set can be smaller than $\omega_1^V$.
Sep 2, 2015 at 12:18 comment added Miha Habič You definitely cannot make a nonrecursive ordinal recursive by forcing. Being recursive is witnessed by a Turing machine, i.e. a natural number, and these aren't affected by forcing.
Sep 2, 2015 at 11:28 history asked Simon Henry CC BY-SA 3.0