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In 1981, building on work by Ellentuck in 1974, Sageev showed ("A model of ZF + there exists an inaccessible, in which the Dedekind cardinals constitute a natural non-standard model of arithmetic," http://www.sciencedirect.com/science/article/pii/0003484381900176) that it is consistent relative to ZF + an inaccessible that the set of Dedekind finite cardinals is linearly ordered - and hence by Ellentuck forms an elementary extension of $\mathbb{N}$.

On page 223 of his paper, Sageev writes:

"In our construction it is necessary to assume that $\kappa$ is inaccessible. It is unknown to the author whether this result is actually bound up with a large cardinal assumption."

My question is:

  • Is Sageev's result known to be equiconsistent with an inaccessible?
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    $\begingroup$ Looking at the journal page of Sageev's paper, there are no mention of any papers citing this one. My guess is that this is either very open, or that there's some private paper sitting in someone's drawer where this is solved. But now you got me intrigued! (Why are his papers so long!! :-() $\endgroup$ – Asaf Karagila Jun 20 '14 at 21:51
  • $\begingroup$ Actually, now that I think about it, Leo Harrington probably knows the answer to this (on the grounds that Leo knows everything); unfortunately, I'm not in Berkeley right now, but if this is still unanswered in a few months, I'll definitely ask him. (And yeah, I'm intrigued too! I wish I were more competent in this sort of problem, it's the sort of thing that in principle I'd love to lose myself in for a time.) $\endgroup$ – Noah Schweber Jun 20 '14 at 21:54
  • $\begingroup$ @AsafKaragila As far as I know, Sageev's papers are long because they solved really hard problems. I don't recall any cases where other people gave simpler proofs of his results. $\endgroup$ – Andreas Blass Jun 20 '14 at 22:00
  • $\begingroup$ @Andreas: Yes, I am aware of that, and I have nothing but respect for him for solving these sort of problems. But I think that his papers are also incredibly long because they were written at an era (or his psyche was still in that era) where a rather complete introduction to forcing was expected in papers about forcing. Strangely, in the paper mentioned here it still seems that he is using the terminology predating Shoenfield's work (even if not the actual concept). This makes his work even harder to decipher from a modern point of view. $\endgroup$ – Asaf Karagila Jun 20 '14 at 22:08

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