Abstract We study descriptive set theory in the space by letting trees with no uncountable branches play a similar role as countable ordinals in traditional descriptive set theory. By using such trees, we get, for example, a covering property for the class of -sets of . We call a family of trees universal for a class of trees if ⊆ and every tree in can be order-preservingly mapped into a tree in . It is well known that the class of countable trees with no infinite branches has a universal family of size ℵ1. We shall study the smallest cardinality of a universal family for the class of trees of cardinality ≤ ℵ1 with no uncountable branches. We prove that this cardinality can be 1 (under ¬CH) and any regular cardinal κ which satisfies (under CH). This bears immediately on the covering property of the -subsets of the space . We also study the possible cardinalities of definable subsets of . We show that the statement that every definable subset of has cardinality <ωn or cardinality is equiconsistent with ZFC (if n ≥ 3) and with ZFC plus an inaccessible (if n = 2). Finally, we define an analogue of the notion of a Borel set for the space and prove a Souslin-Kleene type theorem for this notion.

Abstract It is consistent with ¬ CH that every universal theory of relational structures with the joint embedding property and amalgamation for -(3)-diagrams has a universal model of cardinality ℵ1. For classes with amalgamation for -(4)-diagrams it is consistent that and there is a universal model of cardinality ℵ2.

Abstract It is consistent that for many cardinals λ there is a family of at least λ+ unbounded subsets of λ which have uniformization properties. In particular if it is consistent that a supercompact cardinal exists, then it is consistent that ℵω has such a family. We have applications to point set topology, Whitehead groups and reconstructing separable abelian p-groups from their socles.

Four formalizations of the instruction if–then–else are considered. Let K be an equational class of algebras. Assume the language has constants $ \bot $ in case II, T, F, $ \bot $ in case III and T in case IV. Any equational class with $ \bot $ is assumed to be strict, i.e. satisfies $F(\_, \bot ,\_) = \bot $ for all functions f Form the classes I $K^d $, II $K^{d \bot } $, III $K^t $ and IV $K^{t'} $ by adding an operation $[,,,]$ in cases I and II defined by \[ {\text{I}}\quad [x,y,z,w] = \left\{ \begin{gathered} z\quad {\text{if }}x = y, \hfill \\ w\quad {\text{else}} \hfill \\ \end{gathered} \right.\qquad {\text{II}}\quad [x,y,z,w] = \left\{ \begin{gathered} z\quad {\text{if }}x = y \ne \bot \hfill \\ w\quad {\text{if }} \bot \ne x \ne y \ne \bot , \hfill \\ \bot \quad {\text{else,}} \hfill \\ \end{gathered} \right. \]and by adding an operation $[,,]$ in cases III and IV defined by \[ {\text{III}}\quad [x,y,z] = \left\{ \begin{gathered} y\quad {\text{if }}x = T, \hfill \\ z\quad {\text{if }}x = F \hfill \\ \bot \quad {\text{else}}, \hfill \\ \end{gathered} \right.{\text{ and IV }}[x,y,z] = \left\{ \begin{gathered} y\quad {\text{if }}x = T, \hfill \\ z\quad {\text{else}}{\text{.}} \hfill \\ \end{gathered} \right. \] (In case III only the one point algebra and those algebras in which T, F,$ \bot $ are distinct are used to form $K'$.) In all four cases a finite basis for the equational theory of $K^x (x = d,d \bot ,t,t')$ relative to the equational theory of K are given. For $K^d $ such a basis was previously known. If the word problem for K is decidable then so is the equational theory of $K^x $. For $K^d $, as was previously known, the converse holds, but for $K^t $ the implication does not always reverse.

Abstract c.c.c. posets are characterised in terms of -generic conditions. This characterisation can be applied to get simple proofs of many facts about c.c.c. forcing including Con(MA + ┐CH).

Abstract Let p be an odd prime. A method is described which given a structure M of finite similarity type produces a nilpotent group of class 2 and exponent p which is in the same stability class as M. Theorem. There are nilpotent groups of class 2 and exponent p in all stability classes. Theorem. The problem of characterizing a stability class is equivalent to characterizing the (nilpotent, class 2, exponent p) groups in that class.

The Löwenheim–Skolem theorem states that if a theory has an infinite model it has models of all cardinalities greater than or equal to the cardinality of the language in which the theory is defined. A natural question is what happens if there is a model whose cardinality is less than that of the language. If κ is an infinite cardinal less than the first measurable cardinal and κ < κω, the Rabin–Keiler theorem [1, p. 139] gives an example of a theory which has a model of cardinality κ in which every element is the interpretation of a constant and all other models have cardinality μ ≥ κω. Keisler has also shown that if a theory has a model of cardinality κ it has models of all cardinalities μ ≥ κω. We will show that within the bounds of the above theorems anything can happen. The main result is as follows.