The special Aronszajn tree property

Abstract

Assuming the existence of a proper class of supercompact cardinals, we force a generic extension in which, for every regular cardinal κ, there are κ + -Aronszajn trees, and all such trees are special.

Specializing Trees with Small Approximations I

“…Regrading the consistency results, there are many interesting results. Extending the well-known work of Laver and Shelah [11] regarding the consistency of ℵ 2 -Suslin Hypothesis with CH from a weakly compact cardinal, Golshani and Hayut in [6] proved, modulo the consistency of large cardinals, that it is consistent that for every regular cardinal κ, there are κ + -Aronszajn trees and all of them are special. In a collaboration, Golshani and Shelah [7] proved that it is consistent that every tree of height and size κ + , for a prescribed regular cardinal κ, is special.…”

Club Stationary Reflection and the Special Aronszajn Tree Property

“…Later, Laver and Shelah showed ( [30]) that SATP(ω 2 ) is consistent from a weakly compact cardinal. Generalizing this further, Golshani and Hayut have recently shown ( [19]), using posets which specialize with anticipation, that it is consistent that, simultaneously, for every regular cardinal κ, SATP(κ + ) holds. Krueger has generalized the result of Laver-Shelah (and also Abraham-Shelah, [2]) in a different direction ( [28]), showing that it is consistent that any two countably closed Aronszjan trees on ω 2 are club isomorphic.…”

The Special Aronszajn Tree Property at $κ^+$ and $\square_{κ, 2}$

“…In this section we review the methods of [2] for specializing trees while anticipating subsequent forcing. First we introduce the modified Baumgartner forcing which specializes a single tree while anticipating a single subsequent forcing.…”

“…Golshani and Hayut [2] showed that under the same assumption it is consistent that SATP(ℵ 1 ) and SATP(ℵ 2 ) hold simultaneously, and achieved a global result by showing that it is consistent that SATP(κ + ) holds simultaneously for all regular κ, assuming the existence of a proper class of supercompact cardinals. This result is achieved by adapting the methods of [4] to specialize all possible names for trees of height κ + while anticipating the specialization of trees of height κ.…”

“…For any successor cardinal κ + , we say that a κ + -Aronszajn tree T is special if there exists a function f : T → κ such that if x < T y then f (x) = f (y). Following [2], if there are Aronszajn trees of height κ + and all such trees are special, we say that the Special Aronszajn Tree Property holds at κ + , and denote this by SATP(κ + ).…”