Club isomorphisms on higher Aronszajn trees (Q720758)

In the paper under review, the author proves that it is consistent, relative to an ineffable cardinal, that the continuum hypothesis holds and every two \(\sigma\)-closed \(\omega_2\)-Arsonszajn trees are club isomorphic. Consistency results about the structure of \(\omega_1\)-trees played a crucial role in the development of forcing technique, such as iteration of forcing notions with the countable chain condition [\textit{R. M. Solovay} and \textit{S. Tennenbaum}, Ann. Math. (2) 94, 201--245 (1971; Zbl 0244.02023)] and proper forcing [\textit{S. Shelah}, Proper and improper forcing. 2nd edition. Reprint of the 1998 original published by Springer. Cambridge: Cambridge University Press; Urbana, IL: Association for Symbolic Logic (ASL) (2016; Zbl 1365.03012)]. When trying to generalize consistency results from \(\omega_1\)-trees to \(\omega_2\)-trees one faces the problem that there are no parallel iteration and preservation theorems for \(\omega_2\). Thus, typically, obtaining consistency results for \(\omega_2\)-trees is more difficult, and has non-trivial consistency strength. The result of this paper generalizes two previous results. The first one is from [\textit{U. Abraham} and \textit{S. Shelah}, Isr. J. Math. 50, 75--113 (1985; Zbl 0566.03032)], in which the consistency of club isomorphism between any two \(\omega_1\)-Aronszajn trees is obtained using an iteration of proper forcings. The second one is from [\textit{R. Laver} and \textit{S. Shelah}, Trans. Am. Math. Soc. 264, 411--417 (1981; Zbl 0495.03034)], in which the consistency of ``\(\mathrm{CH}\) and every \(\omega_2\)-Aronszajn tree is special'' is proved, relative to a weakly compact cardinal.