There is a tree property in set-theory and a tree property in model theory. I want the former, not the later.

Definition The tree property holds at some cardinal $\kappa$ if every tree of height $\kappa$ with levels of size less than $\kappa$ has a cofinal branch.

Counterexamples to the tree property at $\kappa$ are called $\kappa$-Aronszajn trees.

In a few instances, I heard people talking about the tree property as a "compactness property". One of the justifications was the following theorem.

Theorem An inaccessible cardinal has the tree property if and only if it is weakly compact.

Since compactness is a very useful tool in model-theory, one would expect that in the absence of compactness, e.g. in infinitary logics, the tree property to be some sort of replacement. Looking around I did not find anything.

So, my question is:

Do you know any usage of the (set-theoretic) tree property in model theory? Even better, do you know any model-theoretic techniques that use the tree property? By technique, I mean something that can be used in various settings, not in only one particular construction.

  • 1
    $\begingroup$ Well, it is equivalent to a weak compactness theorem for $\cal L_{\kappa\kappa}$. $\endgroup$ – Asaf Karagila Mar 10 '17 at 19:16
  • $\begingroup$ @AsafKaragila: You mean for $\kappa$ inaccessible? $\endgroup$ – Ioannis Souldatos Mar 10 '17 at 19:48
  • $\begingroup$ Well, the compactness implies inaccessibility. You can find the proof in Kanamori's "The Higher Infinite". $\endgroup$ – Asaf Karagila Mar 10 '17 at 20:04
  • $\begingroup$ By the theorem I mentioned, the tree property is equivalent to weak compactness for inaccessible $\kappa$. But how about accessible cardinals, even small cardinals like $\aleph_n$, $n\in\omega$? They consistently satisfy the tree property. $\endgroup$ – Ioannis Souldatos Mar 10 '17 at 20:12
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    $\begingroup$ See Enayat's paper "Power like models of set theory" $\endgroup$ – Mohammad Golshani Mar 11 '17 at 7:56

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