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In Exercises 17.17 and 17.18 of Jech's Set Theory book, he shows that if the language $L_\kappa$ satisfies the Weak Compactness Theorem, then $\kappa$ is a weakly inaccessible cardinal. Also, in Theorem 17.13 of the same book, it is shown that if $\kappa$ is strongly inaccessible, then $L_\kappa$ satisfies the Weak Compactness Theorem if and only if $\kappa$ is weakly compact.

This seems to suggest that it is consistent that there is a cardinal $\kappa$ for which $L_\kappa$ satisfies the Weak Compactness Theorem but such that $\kappa$ is only weakly inaccessible and not strongly inaccessible.

Can anyone provide a reference for such a consistency result?

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    $\begingroup$ Have you checked Kanamori? $\endgroup$
    – Asaf Karagila
    Aug 1, 2019 at 7:20
  • $\begingroup$ Assuming a measurable it's consistent that there is a weakly inaccessible $\kappa$ not strongly inaccessible satisfying the tree property (since there is a model having a real-valued measurable cardinal below $2^{\aleph_0}$). But $\kappa^{<\kappa}>\kappa$ in this model, and so the tree property alone is not enough to prove that $\mathcal{L}_{\kappa, \kappa}$ has the compactness property. $\endgroup$
    – godelian
    Aug 1, 2019 at 14:23
  • $\begingroup$ @godelian Sorry, but I don’t see how that answers the question. $\endgroup$
    – Isaac
    Aug 1, 2019 at 16:38
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    $\begingroup$ due to Solovay and Kunen unpublished (I think in $Add(\omega, \kappa)$ $\kappa$ should have the weak compactness property) but Williams Boos has an article extending their results (my internest sucks now but the title may be something like: weak compactness theorem without strong inaccessibility). $\endgroup$
    – Jing Zhang
    Aug 1, 2019 at 17:08
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    $\begingroup$ Theorem 1.3 of that paper gives it. It says after $Add(\omega, \mu)$ where $\mu$ is a measurable cardinal, in $V[G]$ there exists $\kappa<\mu=2^\omega$ that satisfies the Weak Compactness Theorem. Such cardinal can't be strongly inaccessible obviously. $\endgroup$
    – Jing Zhang
    Aug 1, 2019 at 21:59

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