Logical theories of trees

• Main Author: Kellerman, Ruaan
• Format: Others
• Language: en
• Published: 2011
• Online Access: http://hdl.handle.net/10539/9150

Summary Trees occur naturally in many mathematical settings as important partial orders yet no systematic study of their rst-order theories exists. We investigate some of the rst-order theories of trees. The two problems which motivate the thesis are (i) the rst-order de nability of sets within a given tree, and (ii) the rst-order de nability and axiomatisability of particular classes of trees. Of particular interest is the correspondence between the rst-order theory of a tree and the rst-order theory of the class of linear orders which comprise the paths in the tree. For every class C of linear orders we introduce eight classes of trees collectively called the C-classes of trees, the paths of which are related in various natural ways to the linear orders in C. We completely establish both the set-theoretical relationships between these eight classes of trees as well as the relationships between the rst-order theories of these eight classes of trees. We also investigate some of the properties of these rst-order theories. A special case is where the class C consists of a single ordinal with < !! since such ordinals are nitely axiomatisable. We obtain the rstorder theory of the class of trees where every path is isomorphic to the ordinal for any nite ordinal and also for the case where = !. The remaining cases are more di cult because of the existence of unde nable paths in the tree. For the cases where ! < < !! we introduce the notion of an almost -tree and show that every almost -tree can be elementarily extended in a natural way to a tree of which every path, de nable or unde nable, satis es the rst-order theory of . We also examine what this elementary extension of the almost -tree looks like for the case where = ! + 1. Throughout the thesis we also investigate various rst-order properties and theories of trees and establish some results in this regard.