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Motivated by this question. For us "$\kappa$ has tree property" means any tree that has underlying set $\kappa$, height $\kappa$ and levels $<\kappa$ has a branch.

The strength of $\mathsf{ZF+}$ "$\omega_1$ has tree property" is exactly that of a weakly compact; the forcing direction is the same construction as Jech's making $\omega_1$ measurable. But what if we consider $L(\mathbb{R})$? Feel free to add $\mathsf{DC}$ if that matters.

More broadly, is there some statement $\varphi$ such that the strength of $L(\mathbb{R})\models\varphi$ is exactly a weakly compact? Note that

$\varphi=$ there exists a countable transitive model of $\mathsf{ZFC+}$ "there is a weakly compact cardinal"

does not work, and is cheating anyway. If there turns out to be some other way of cheating, my next question would be whether there is a natural statement $\varphi$.

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    $\begingroup$ There are over a dozen common characterizations of weak compactness in ZFC, but they are not all equivalent in ZF+DC, so could you let us know which version of weakly compact you want? $\endgroup$
    – Joel David Hamkins
    Commented 9 hours ago
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    $\begingroup$ @JoelDavidHamkins Thanks for reminding me. I chose to use the definition via tree property and added the requirement that the underlying set should be $\kappa$, since otherwise I don't see how measurability implies weak compactness. Is it necessary? $\endgroup$
    – new account
    Commented 9 hours ago
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    $\begingroup$ But the tree property alone is not equivalent to weak compactness even in ZFC, unless one also specifies that the cardinal is inaccessible. But $\omega_1$ is never inaccessible. $\endgroup$
    – Joel David Hamkins
    Commented 9 hours ago
  • $\begingroup$ The definition Jech uses in his paper in which he proves $\omega_1$ can be weakly compact or measurable (each relative to such a large cardinal in ZFC) is the partition characterization: an uncountable well-ordered cardinal $\kappa$ is weakly compact if $\kappa \rightarrow (\kappa)^2_2.$ $\endgroup$
    – Elliot Glazer
    Commented 8 hours ago

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Suppose $V=L$ and $\kappa$ is weakly compact. Force with the Levy collapse below $\kappa$. Then in $M=L(\mathbb{R}^{L[G]})$, $\omega_1$ has the tree property. For if $T\in M$ is a tree of the relevant kind, then $T\in\mathrm{HOD}^{L[G]}_{\{x\}}$ for some $x\in\mathbb{R}^{L[G]}$ (since $M=L(\mathbb{R})^{L[G]}$). Let $\alpha<\kappa$ be such that $x\in L[G\upharpoonright\alpha]$. Then $T\in L[G\upharpoonright\alpha]$, by homogeneity of the tail of the Levy collapse. Since $\kappa$ is weakly compact in $L[G\upharpoonright\alpha]$, and $T$ also has the right form in $L[G\upharpoonright\alpha]$, there is $b\in L[G\upharpoonright\alpha]$ which is a $T$-cofinal branch. But then $L[G\upharpoonright\alpha]\subseteq M$, so $b\in M$. Note we also have DC in $M$.

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    $\begingroup$ Thanks a lot. I should have tried this before posting the question...Is there a "metatheorem" on the sort of large cardinal properties that can be transferred to $\omega_1$? According to this answer, Jech's construction works for any large cardinal property that is preserved under small forcing and is of the form $\forall X\exists Y\ \phi(X,Y)$, where $\phi$ is something upward absolute. Your argument seems to work in this generality too? $\endgroup$
    – new account
    Commented 8 hours ago
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    $\begingroup$ Yes, it does seem to work in that generality. $\endgroup$
    – Farmer S
    Commented 8 hours ago

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