[bibtex key=CodyGitikHamkinsSchanker2015:LeastWeaklyCompact]
Abstract. We prove from suitable large cardinal hypotheses that the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly $\theta$-supercompact, for any desired $\theta$. In addition, we prove several global results showing how the entire class of weakly compact cardinals, a proper class, can be made to coincide with the class of unfoldable cardinals, with the class of weakly measurable cardinals or with the class of nearly $\theta_\kappa$-supercompact cardinals $\kappa$, for nearly any desired function $\kappa\mapsto\theta_\kappa$. These results answer several questions that had been open in the literature and extend to these large cardinals the identity-crises phenomenon, first identified by Magidor with the strongly compact cardinals.
In this article, we prove that the least weakly compact cardinal can exhibit any of several much stronger large cardinal properties. Namely, the least weakly compact cardinal can be unfoldable, weakly measurable and nearly $\theta$-supercompact for any desired $\theta$.
Main Theorem. Assuming a suitable large cardinal hypothesis, the least weakly compact cardinal can be unfoldable, weakly measurable and even nearly $\theta$-supercompact, for any desired $\theta$.
Meanwhile, the least weakly compact cardinal can never exhibit these extra large cardinal properties in $L$, and indeed, the existence of a weakly measurable cardinal in the constructible universe is impossible. Furthermore, in each case the extra properties are strictly stronger than weak compactness in consistency strength.
We show in addition a more global result, that the entire class of weakly compact cardinals can be made to coincide with the class of unfoldable cardinals, with the class of weakly measurable cardinals, and with the class of nearly $\theta_\kappa$-supercompact cardinals $\kappa$, with enormous flexibility in the map $\kappa\mapsto\theta_\kappa$.
Our results therefore extend the `identity-crises’ phenomenon—first identified (and named) by Magidor—which occurs when a given large cardinal property can be made in various models to coincide either with much stronger or with much weaker large cardinal notions. Magidor had proved that the least strongly compact cardinal can be the least supercompact cardinal in one model of set theory and the least measurable cardinal in another. Here, we extend the phenomenon to weak measurability, partial near supercompactness and unfoldability. Specifically, the least weakly measurable cardinal coincides with the least measurable cardinal under the GCH, but it is the least weakly compact cardinal in our main theorem. Similarly, the least cardinal $\kappa$ that is nearly $\kappa^{+}$-supercompact is measurable with nontrivial Mitchell order under the GCH, but it is the least weakly compact cardinal here (and similar remarks apply to near $\kappa^{++}$-supercompactness and so on). The least unfoldable cardinal is strongly unfoldable in $L$, and therefore a $\Sigma_2$-reflecting limit of weakly compact cardinals there, but it is the least weakly compact cardinal in our main theorem. The global results of section 6 show just how malleable these notions are.
I am currently reading through this paper and have a question. Since Kunen was able to prove that “It is relatively consistent with ZFC that a cardinal k [for kappa] is not weakly compact, but becomes weakly compact and indeed much more (measurable, strong, strongly compact, supercompact) in a forcing extension V[G] obtained by forcing with a certain k-Souslin tree”, is there some reason that the least weakly compact cardinal cannot be forced to be supercompact? I ask the question because Olivier Esser was able to prove that his system GPK^(+)_Infinity is equconsistent with KM + ‘On is a weakly compact cardinal’ and GPK^(+)_Infinity is supposed to be the system that recovers “the whole theory of Frege: any formula defines a set the Russell’s paradox being tolerated (“A Strong Model od Paraconsistent Logic”, Notre Dame Journal of Formal Logic Vol 44, Number 3, 2003.) Since it is known that KM is able to define models of ZFC (and by extension outer models of ZFC as well), could one use KM + ‘On is a weakly compact cardinal’ as a forcing language not only for the results in this paper, but for the result that the entire class of weakly compact cardinals can be made to coincide with the class of supercompact cardinals?
I am currently reading your paper and want to know if there is any reason the results in this paper cannot be extended to show that the class of weakly compact cardinals can be made to coincide with the class of supercompact cardinals. After all, Kunen showed (in a theorem you quote) that a cardinal k (for kappa) not weakly compact can become supercompact in a forcing extension V[G] obtained by forcing with a certain k-Souslin tree. I am asking this question because Olivier Esser has shown that his system GPK^(+)_Infinity is equiconsistent with KM + ‘On is a weakly compact cardinal’ and this system was supposed to “recover the whole theory of Frege: any formula defines a set, the Russell’s paradox being tolerated” ( ” A Strong System of Paraconsistent Logic”, Notre Dame Journal of formal Logic, Vol. 44, Number 3, 2003. It seems to me that a system that claims to recover “the whole theory of Frege” should not have a ‘cut-off’ point at a weakly compact cardinal but should be able to formulate supercompactness and notions of all larger cardinals since it is supposed to be a ‘paraconsistent’ theory of sets.
Every supercompact cardinal and indeed every measurable cardinal is a limit of weakly compact cardinals, and so it is not possible for the class of weakly compact cardinals to coincide with the supercompact cardinals. But your other question, about whether the least weakly compact cardinal can become supercompact in a forcing extension, is quite interesting, and I want to think more about it. Of course, the resurrection forcing would have to revive many smaller large cardinals as well.