We prove that every model of $T = \mathrm{Th}(\omega, countable) has an end extension; and that every countable theory with an infinite order and Skolem functions has 2 ℵ 0 nonisomorphic countable models; and that if every model of T has an end extension, then every |T|-universal model of T has an end extension definable with parameters.

Suppose A is a linear order, possibly with countably many unary predicates added. We classify the isomorphism relation for countable models of Th\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {Th}$$\end{document} up to Borel bi-reducibility, showing there are exactly five possibilities and characterizing exactly when each can occur in simple model-theoretic terms. We show that if the language is finite, then the theory is ℵ0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\aleph _0$$\end{document}-categorical or Borel complete; this (...) generalizes a theorem due to Schirmann. (shrink)

We prove that in $\operatorname {NTP}_{\operatorname {2}}$ theories the dp-rank of a type can be witnessed by indiscernible sequences of tuples satisfying that type. If the type has dp-rank infinity, then this can be witnessed by singletons.

0-categorical o-minimal structures were completely described by Pillay and Steinhorn 565–592), and are essentially built up from copies of the rationals as an ordered set by ‘cutting and copying’. Here we investigate the possible structures which an 0-categorical weakly o-minimal set may carry, and find that there are some rather more interesting examples. We show that even here the possibilities are limited. We subdivide our study into the following principal cases: the structure is 1-indiscernible, in which case all possibilities are (...) classified up to binary structure; the structure is 2-indiscernible, classified up to ternary structure; the structure is 3-indiscernible, in which case we show that it is k-indiscernible for every finite k. We also make some remarks about the possible structures of higher arities which an 0-categorical weakly o-minimal structure may carry. (shrink)