We show that club isomorphism of Kurepa trees have consequences about large cardinals. More precisely, assume that every two Kurepa trees are club isomorphic. Then $\omega_2$ is inaccessible in the constructible universe $\textbf{L}$.
Large Cardinals and Definable Well-Orderings of the Universe
Model-theoretic Characterizations of Large Cardinals
Forcing "$\mathrm{NS}_{ω_1}$ is $ω_1$-dense" From Large Cardinals
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