Ultraproducts of O-Minimal Structures
- Author(s): Rennet, Alex
- Advisor(s): Scanlon, Thomas
- et al.
There are three main parts to this thesis, all centred around ultraproducts of o-minimal structures.
In the first part we investigate (for a fixed first-order language L) what we call the L-theory of o-minimality. It is the theory consisting of those L-sentences true in all o-minimal L-structures. We find that when L expands the language of real closed fields by at least one new function or relation symbol, the L-theory of o-minimality is not recursively axiomatizable. In particular, for any recursive list of
axioms A which is consistent with the L-theory of o-minimality, we find that there are locally o-minimal, definably complete structures satisfying A which are not elementarily equivalent to an ultraproduct of o-minimal structures. We call the latter sort of structures pseudo-o-minimal.
In the second part we investigate uniform finiteness and cell decomposition in the pseudo-o-minimal setting. To do this, we introduce the notion of a pseudo-o-minimal structure tallying a discrete definable set. Investigating this notion, we answer some questions of uniqueness and existence. Finally, we show that under certain assumptions about the discrete definable sets that a given pseudo-o-minimal structure can tally, we have a version of uniform finiteness, at least in the planar case. This is the first step towards a cell decomposition theorem in this setting.
In the final section, we look into two classes of examples of ultraproducts of o-minimal structures. For the first class, we note the o-minimality of a certain subset of these structures, and show the non-o-minimality of another. In particular, we derive the o-minimality of a new structure related to the real field with the exponential function. The second class is relatively intractable, but we discuss its relation to an important open problem in o-minimality.