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Mar 22, 2020 at 15:35 comment added Todd Eisworth @Rahman.M I'm still trying to map out what is known and unknown, so I should have more questions on here soon. Shelah has some ZFC limits on what can happen larger cardinals.
Mar 22, 2020 at 10:10 comment added Rahman. M I had asked Boban about this question and I think it is still open. I like this problem very much and it is really interesting. It can serve as a test question for finding the method of size conditions with finite conditions which can add objects of size $\aleph_3$. I don't remember if Boban has mentioned any result around this problem to me. I will ask him in the next online meeting!
Mar 22, 2020 at 2:42 history asked Todd Eisworth CC BY-SA 4.0