Olzhas Umbetbayev (Institute of Mathematics and Mathematical Modeling, Almaty)

Non-essential expansion of small theory and number of countable non-isomorphic models

Note date and time change: Thu 1.30pm

Authors: Bektur BAIZHANOV, Olzhas UMBETBAYEV, Tatyana ZAMBARNAYA

A. Let $T$ be a small theory, $p\in S(T)$ be a non-isolated type, $T_1:= T\cup p(\bar c)$ be a non-essential expansion. There are examples of small theories 1-2 such that:

1. $I(T, \omega)< \omega$, $I(T_1, \omega) = \omega$. (R. Woodrow, M.Peretyat'kin).
2. $I(T,\omega)= \omega$, $I(T_1, \omega)< \omega$ (B. Omarov)
3. Question. Is there a small theory such that $I(T, \omega) =2^{\omega}$, $I(T,\omega)\leq \omega$?

We discuss the usage of constant expansion on research of Vaught Conjecture.

B. Let $\mathfrak M$ be a model of a theory $T$. A finite diagram (S. Shelah) of $\mathfrak M$ is the collection of all $\emptyset$-definable complete types that are realized in $\mathfrak M$:

$$\mathcal D(\mathfrak M)=\{p\in S(T)\ |\ \mathfrak M\models p\}.$$

A dowry of a set $B\subseteq M$ is the following collection:

$$\mathcal D(B)=\{p\in S(T)\ | \text{ there exist }\mathfrak M\models T \text{ and } \bar b\in B \text{ such that } \bar b\in p(\mathfrak M)\}.$$

Conjecture. If $|\{ \mathcal D(\mathfrak M) \mid \mathfrak M\models T\}| >\omega$, then $|\{ \mathcal D(\mathfrak M) \mid \mathfrak M\models T\}| =2^{\omega}$, and, consequently, $I(T,\omega)= 2^{\omega}$.

Question. Let $P:=\{p_n\in S(T) \mid n<\omega\}$, $Q:=\{q_n\in S(T) \mid n <\omega\}$ be two families of non-isolated types such that for every $n<\omega$ there exists $\mathfrak M_n$ is realized first $n$ types from $P$ and is omitted first $n$ types from $Q$. Do there exist a countable model $\mathfrak M$ of $T$, such that it realizes each type from $P$ and omits each type from $Q$?

The criterion for the existence of such a theory will be shown.