It is proved that the following conditions are equivalent:

(a) there exists a complete, atomless, σ–centered Boolean algebra, which does not contain any regular, atomless, countable subalgebra.

(b) there exists a nowhere dense ultrafilter on ω.

Therefore, the existence of such algebras is undecidable in ZFC. In “forcing language” condition (a) says that there exists a non–trivial σ–centered forcing not adding Cohen reals.

We present two ways in which the model L(ℝ) is canonical assuming the existence of large cardinals. We show that the theory of this model, with ordinal parameters, cannot be changed by small forcing: we show further that a set of ordinals in V cannot be added to L(ℝ) by small forcing. The large cardinal needed corresponds to the consistency strength of ADL(ℝ): roughly ω Woodin cardinals.

It is shown that for any cardinal κ, , and if κ is weakly compact then .

Firstly, in this paper, we prove that the equivalence of simplicity and the symmetry of forking. Secondly, we attempt to recover definability part of stability theory to simplicity theory. In particular, using elimination of hyperimaginaries we prove that for any supersimple T. canonical base of an amalgamation class is the union of names of ψ-definitions of , ψ ranging over stationary L-formulas in . Also, we prove that the same is true with stable formulas for an 1-based theory having elimination of hyperimaginaries. For such a theory, the stable forking property holds, too.

We will study complete Ln-theories and their models, where Ln is the set of first order formulas in which at most n distinct variables occur. Here, by a complete Ln-theory we mean a theory such that for every Ln-sentence, it or its negation is implied by the theory. Hence, a complete Ln-theory need not necessarily be complete in the usual sense. Our approach is to transfer concepts and methods from stability theory, such as the order property and counting types, to the context of Ln-theories. So, in one sense, we will develop some rudimentary stability theory for a particular class of (possibly) incomplete theories. To make the ‘stability theoretic’ arguments work, we need to assume that models of the complete Ln-theory T which we consider can be amalgamated in certain ways. If this condition is satisfied and T has infinite models then there will exist models of T which are sufficiently saturated with respect to Ln. This allows us to use some counting types arguments from stability theory. If, moreover, we impose some finiteness conditions on the number of Ln-types and the length of Ln-definable orders then a sufficiently saturated model of T will be ω-categorical and ω-stable. Using the theory of ω-categorical and ω-stable structures we derive that T has arbitrarily large finite models.

A different approach to combining stability theory with finite model theory is made by Hyttinen in [9] and [10].

We propose a model-theoretic framework for investigating profinite groups. Within this framework we define and investigate small profinite groups. We consider the question if any small profinite group has an open abelian subgroup.

It is shown that for each computably enumerable set of n-element subsets of ω there is an infinite set A ⊆ ω such that either all n-element subsets of A are in or no n-element subsets of A are in . An analogous result is obtained with the requirement that A be replaced by the requirement that the jump of A be computable from 0(n). These results are best possible in various senses.

We show there is a non-recursive r.e. set A such that if W is any low r.e. set. then the join W ⊕ A is also low. That is. A is “almost deep”. This answers a question of Joekusch. The almost deep degrees form an definable ideal in the r.e. degrees (with jump.)

Strengthening known instances of Vaught Conjecture, we prove the Glimm-Effros dichotomy theorems for countable linear orderings and for simple trees. Corollaries of the theorems answer some open questions of Friedman and Stanley in an Lω1ω-interpretability theory. We also give a survey of this theory.

We show that every infinite computable partial ordering has either an infinite chain or an infinite antichain. Our main result is that this cannot be improved: We construct an infinite computable partial ordering that has neither an infinite chain nor an infinite antichain.

In this article we introduce systems for metapredicative Mahlo in explicit mathematics and admissible set theory. The exact upper proof-theoretic bounds of these systems are established.

Answers are given to two questions concerning the existence of some sparse subsets of ℋ = {0, 1 … ., H – 1} ⊆ *ℕ. where H is a hyperfinite integer. In §1. we answer a question of Kanovei by showing that for a given cut U in ℋ, there exists a countably determined set X ⊆ ℋ which contains exactly one element in each U-monad, if and only if U = a · ℕ for some a Є ℋ ∖ {0}. In §2, we deal with a question of Keisler and Leth in [6]. We show that there is a cut V ⊆ ℋ such that for any cut U, (i) there exists a U-discrete set X ⊆ ℋ with X + X = ℋ (mod H) provided , (ii) there does not exist any U-discrete set X ⊆ ℋ with X + X = ℋ (mod H) provided . We obtain some partial results for the case U = V.