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In the paper, "Forcing with sequences of models of two types," (MR3201836), Neeman claims that, using a supercompact and a weakly compact above, one can force with his pure side conditions poset twice to obtain the tree property at $\omega_2$ and $\omega_3$.

Question 1: Are the details of this written up anywhere?

Question 2: Does the second stage add reals? (For a fairly simple reason?)

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  • $\begingroup$ It is probably implicit in worldscientific.com/doi/10.1142/S0219061321500033. It contains an extension of that result. $\endgroup$
    – Otto
    Commented Oct 17, 2020 at 13:02
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    $\begingroup$ @Otto I think the Mohammapour-Velickovic forcing is quite different. They do not iterate like in Neeman’s remark. One main difference: if $\theta<\theta’$ are the cardinals that become $\omega_2$ and $\omega_3$ in either scenario, only $\theta$-many countable models show up in the generic of Neeman’s two-step iteration. $\endgroup$
    – Monroe Eskew
    Commented Oct 17, 2020 at 13:04
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    $\begingroup$ It would be nice to have a general update on the status of Neeman's work on higher analogues of proper forcing. $\endgroup$
    – Erik Walsberg
    Commented Oct 18, 2020 at 0:30

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I believe that Neeman’s claim was in error. This preprint contains a kind of counterexample.

Version 1 of the paper, available under the link on the arxiv, contains a detailed attempt to prove Neeman‘s claim based on my discussions with him. After writing it all up, I thought about the argument again and started to worry that there were some gaps/circularities in the proof of the main technical lemma, Lemma 32, and a mistake in the argument for the tree property at $\kappa$ near the end. (The copy of $\omega^{<\omega}$ I built isn’t dense.) After struggling to correct it, I gave up and started thinking in the other direction.

His claim included an answer to question 2 in the negative; the second stage should not add reals. My counterexample is to the published claim conjoined with that unpublished subclaim. But maybe a completely different argument from what Neeman told me may prove the original claim. The question is, what is that “special argument” for preserving $\omega_1$?

I told Neeman about this work, and he said that a modification of the forcing that directly avoids my counterexample should work. (A Laver function is incorporated to bound the ordertypes of Magidor models.) He says he will write it up.

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