In [7] it is shown that if Σ is a type omitted in the structure = ω, +, ·, < and complete with respect to Th() then Σ is omitted in models of Th() of all infinite powers. The proof given there extends readily to other models of P. In this paper the result is extended to models of ZFC. For pre-tidy models of ZFC, the proof is a straightforward combination of the methods in [7] and in Keisler and Morley ([9], [6]). For other models, the proof involves forcing. In particular, it uses Solovay and Cohen's original forcing proof that GB is a conservative extension of ZFC (see [2, p. 105] and [5, p. 77]).

The method of proof used for pre-tidy models of set theory can be used to obtain an alternate proof of the result for This new proof yields more information. First of all, a condition is obtained which resembles the hypothesis of the “Omitting Types” theorem, and which is sufficient for a theory T to have a model omitting a type Σ and containing an infinite set of indiscernibles. The proof that this condition is sufficient is essentially contained in Morley's proof [9] that the Hanf number for omitting types is so the condition will be called Morley's condition.

If T is a pre-tidy theory, Morley's condition guarantees that T will have models omitting Σ in all infinite powers.

We generalise various properties of quasiendomorphisms from groups with regular generic to small abelian groups. In particular, for a small stable abelian group such that no infinite definable quotient is connected-by-finite, the ring of quasi-endomorphisms is locally finite. Under some additional assumptions, it decomposes modulo some nil ideal into a sum of finitely many matrix rings.

We discuss saturating ultrafilters on N, relating them to other types of non-principal ultrafilter. (a) There is an (ω, c)-saturating ultrafllter on N iff 2λ ≤ c for every λ < c and there is no cover of R by fewer than c nowhere dense sets, (b) Assume Martin's axiom. Then, for any cardinal κ, a nonprincipal ultrafllter on N is (ω, κ)-saturating iff it is almost κ-good. In particular, (i) p(κ)-point ultrafilters are (ω, κ)-saturating, and (ii) the set of (ω, κ)-saturating ultrafilters is invariant under homeomorphisms of βN/N. (c) It is relatively consistent with ZFC to suppose that there is a Ramsey p(c)-point ultrafilter which is not (ω, c)-saturating.

There is a PCA well-ordering of the real line if and only if there is a real from which every real is constructible.

We present some abstract theorems showing how domination properties equivalent to being or array nonrecursive can be used to construct sets generic for different notions of forcing. These theorems are then applied to give simple proofs of some known results. We also give a direct uniform proof of a recent result of Ambos-Spies, Ding, Wang, and Yu [2009] that every degree above any in is recursively enumerable in a 1-generic degree strictly below it. Our major new result is that every array nonrecursive degree is r.e. in some degree strictly below it. Our analysis of array nonrecursiveness and construction of generic sequences below ANR degrees also reveal a new level of uniformity in these types of results.

This article is concerned with functions k assigning a cardinal number to each infinite Boolean algebra (BA), and the behaviour of such functions under ultraproducts. For some common functions k we have

for others we have ≤ instead, under suitable assumptions. For the function π character we go into more detail. More specifically, ≥ holds when F is regular, for cellularity, length, irredundance, spread, and incomparability. ≤ holds for π. ≥ holds under GCH for F regular, for depth, π, πχ, χ, h-cof, tightness, hL, and hd. These results show that ≥ can consistently hold in ZFC since if V = L holds then all uniform ultrafilters are regular. For π-character we prove two more results: (1) If F is regular and ess , then

(2) It is relatively consistent to have , where A is the denumerable atomless BA.

A thorough analysis of what happens without the assumption that F is regular can be found in Rosłanowski, Shelah [8] and Magidor, Shelah [5]. Those papers also mention open problems concerning the above two possible inequalities.

Given a metric space (X , d), equipped with a locally finite Borel measure, a measurable set $A \subseteq X$ is a density set if the points where A has density 1 are exactly the points of A. We study the topological complexity of the density sets of the real line with Lebesgue measure, with the tools—and from the point of view—of descriptive set theory. In this context a density set is always in $\Pi _3^0$. We single out a family of true $\Pi _3^0$ density sets, an example of true $\Sigma _2^0$ density set and finally one of true $\Pi _2^0$ density set.

We study the relative computational power of structures related to the ordered field of reals, specifically using the notion of generic Muchnik reducibility. We show that any expansion of the reals by a continuous function has no more computing power than the reals, answering a question of Igusa, Knight, and Schweber [7]. On the other hand, we show that there is a certain Borel expansion of the reals that is strictly more powerful than the reals and such that any Borel quotient of the reals reduces to it.

Orthogonality between two stably embedded definable sets is preserved under the addition of constants.