Abstract
The paper reviews special Aronszajn trees, both at ω1 and κ+ for an uncountable regular κ. It provides a comprehensive classification of the trees and discusses the existence of these trees under different set-theoretical assumptions. The paper provides details and proofs for many folklore results which circulate (often without a proper proof) in the literature.
Highlights
A tree, which is called Aronszajn, was first constructed by Nachman Aronszajn and the construction can be found in [Kur35]
The first one leads to the notion of a special Aronszajn tree, to which we dedicate the section
We introduce the concept of an M-special Aronszajn tree
Summary
A tree, which is called Aronszajn, was first constructed by Nachman Aronszajn and the construction can be found in [Kur35]. The constructed tree was a special ω1-Aronszajn tree. The definition of special Aronszajn tree has several equivalent variants and in the literature can be found many generalizations of the definition of a special Aronszajn tree. In this paper we focus on the question what are the relationships between them and provide a basic classification
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