# The PP Conjecture in the Theory of Spaces of Orderings

Is it true that if a positive primitive formula holds in every finite subspace of a space of orderings, then it also holds in the whole space?

This problem is now known as the pp conjecture. The answer to this question is affirmative in many cases, although it has always seemed unlikely that the conjecture has a positive solution in general. In this thesis, we discuss, discovered by us, first counterexamples for which the pp conjecture fails. Namely, we classify spaces of orderings of function fields of rational conics with respect to the pp conjecture, and show for which of such spaces the conjecture fails, and then we disprove the pp conjecture for the space of orderings of the field R(x,y). Some other examples, which can be easily obtained from the developed theory, are also given. In addition, we provide a refinement of the result previously obtained by Vincent Astier and Markus Tressl, which shows that a pp formula fails on a finite subspace of a space of orderings, if and only if a certain family of formulae is verified.

Advisor:Marshall, Murray

School:University of Saskatchewan

School Location:Canada - Saskatchewan

Source Type:Master's Thesis

Keywords:spaces of orderings forms over real fields

ISBN:

Date of Publication:09/18/2007