On automorphisms of structures in logic and orderability of groups in topology.

摘要

We investigate properties of non-standard models of set theory, in particular, the countable recursively saturated models while having the automorphisms of such models in mind. The set of automorphisms of a model forms a group that in certain circumstances can give information about the model and even recover the model. We develop results on conditions for the existence of automorphisms that fix a given initial segment of a countable recursively saturated model of ZF.;In certain cases an additional axiom V = OD will be needed in order to establish analogues of some results for models of Peano Arithmetic. This axiom will provide us with a definable global well-ordering of the model. Models of set theory do not automatically possess such a well-ordering, but a definable well-ordering is already in place for models of Peano Arithmetic, that is, the natural order of the model.;We investigate some finitely presentable groups that arise from topology. These groups are the fundamental groups of certain manifolds and their orderability properties have implications for the manifolds they come from. A group ⟨G, ∘⟩ is called left-orderable if there is a total order relation < on G that preserves the group operation ∘ from the left.;Finitely presentable groups constitute an important class of finitely generated groups and we establish criteria for a finitely presented group to be non-left-orderable. We also investigate the orderability properties for Fibonacci groups and their generalizations.