Professor John Krueger invites you to attend the

Doctoral Dissertation Defense of Jill Kaiser

"Club Isomorphisms Between Subtrees of Aronszajn Trees"

ABSTRACT:

Aronszajn trees were first introduced by Nachman Aronszajn in the 1930s. Two distinct types of Aronszajn trees are Suslin trees and special Aronszajn trees; no Aronszajn tree can be both special and Suslin. Special Aronszajn trees exist from ZFC. However, as proven by Solovay and Tennenbaum in 1971, the existence of a Suslin tree is independent of ZFC. Assuming ♢, we get that there exists a Suslin tree (Jensen), and we get that there exists 2ℵ1 many non-isomorphic Aronszajn trees such that none of those Aronszajn trees are special or are Suslin (Devlin).

In 1985, Abraham and Shelah proved that 2^ℵ0 < 2^ℵ1 implies that there are 2^ℵ1 many non-club isomorphic Aronszajn trees. In 2022, Krueger showed that it is consistent with ZFC to have 2^ℵ0 = ℵ₂, there exists a Suslin tree, and every pair of normal Aronszajn trees, such that neither contain a Suslin subtree, are club isomorphic to each other.

In this defense, we will discuss the result that it is consistent with ZFC that GCH holds and that every pair of normal Aronszajn trees contain subtrees which are club isomorphic to each other. Since special Aronszajn trees exist from ZFC, when every pair of normal Aronszajn trees contain club isomorphic subtrees, then there are no Suslin trees.