The first order theory of a dense pair and a discrete group

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Abstract

Let $\tilde{T}$ be the theory of an expansion of $\la \bar{\mathbb{R}},+,.,0,1,<\ra$ which is o-minimal, model complete and polynomially bounded with $\mathbb{Q}$-exponents. We introduce a theory $\mathbb{T}$ whose models are of the form $\mathbb{M}=(\tilde{M},G,A)$, where $\tilde{M},\tilde{G}\models \tilde{T}$, $\tilde{G}$ is an elementary substructure of $\tilde{M}$, $G$ is dense in $M$, and $A$ is a discrete multiplicative subgroup of $(M,.)$.We will prove that $\mathbb{T}$ is complete and hence it axiomatises $\Th((\bar{\mathbb{R}},\mathbb{R}_{alg},2^{\mathbb{Z}}))$ when $\tilde{T}$ is $\Th(\bar{\mathbb{R}})$. We will then prove that if $\mathbb{M}\models \mathbb{T}$ and $\psi(\bar{z})$ is a formula in $L(\mathbb{T})(M)$, then it has an equivalent which is a Boolean combination of the formulas of the form \[\exists \bar{x}\in G \ \ \exists \bar{y}\in A\quad \phi(\bar{x},\bar{y}, \bar{z}) \]where $\phi(\bar{x},\bar{y},\bar{z})$ is a formula in $L(\tilde{T})(M)$. Using this, we will characterise the definable sets and the types of tuples in $M$, for a model $\mathbb{M}$ of $\mathbb{T}$. This characterisation, says in particular that $(\mathbb{Z},+,.,<)$ is not interpretable in a model of $\mathbb{T}$ and in spite of having a discrete and a dense subset in our structures, they are tame regarding the fact they do not exhibit the G\"{o}del phenomenon. \par We will note that the open definable subsets of $M$ in $( \tilde{M},G,A)$ can be defined in $( \tilde{M},A)$, and towards proving that every open definable subset of $\mathbb{R}^n$ in $(\bar{\mathbb{R}},\mathbb{R}_{alg},2^{\mathbb{Z}})$ is definable in $(\bar{\mathbb{R}},2^{\mathbb{Z}})$, we will prove that open definable subsets of $\mathbb{R}^n$ which are defined by \emph{special} formulas with parameters in $\mathbb{R}_{alg}$ can be defined in $(\bar{\mathbb{R}},2^{\mathbb{Z}})$.In our last chapter, we will prove that $\mathbb{T}$ has NIP (not the independence property).

Layman's Abstract

In this thesis we have shown that a seemingly complicated mathematical structure can exhibit "tame behaviour". The structure we have dealt with is a field (a space in which there are addition and multiplication which satisfy natural properties) together with a dense subset (a subset which has spread in all parts of the this set, as Q does in R) and a discrete subset (a subset comprised of single points which keep certain distances from one another). This tameness is essentially with regards to not being trapped with the “Godel phenomeono” as the Peano arithmetic does.

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Keyword(s)

Godel phenomenon, dense pairs, discrete sets, tameness, o-minimal

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