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Oct 4, 2016 at 16:30 comment added Vincenzo Dimonte Yes, unfortunately the paper on generic $I_0$ does not give a forcing construction as it is asked in the question, it just says IF there is a forcing construction with such and such properties THEN there are consequences on small cardinals so specific that it would be weird if they do not come from $I_0$. For what is worth, I don't know any attempt of proving even just generic 3-huge, and I don't know any forcing construction that uses $I_0$ in a meaningful way.
Sep 28, 2016 at 2:43 history edited François G. Dorais CC BY-SA 3.0
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Sep 27, 2016 at 3:20 history answered Mohammad Golshani CC BY-SA 3.0