In this paper we will present a simple condition for an ideal to be nowhere precipitous. Through this condition we show nowhere precipitousness of fundamental ideals on Pkλ, in particular the non-stationary ideal NSkλ under cardinal arithmetic assumptions.

In this section I denotes a non-principal ideal on an infinite set A. Let I+ = PA / I (ordered by inclusion as a forcing notion) and I∣X = {Y ⊂ A: Y ⋂ X ∈ I}, which is also an ideal on A for X ∈ I+. We refer the reader to [8, Section 35] for the general theory of generic ultrapowers associated with an ideal. We recall I is said to be precipitous if ⊨I+ “Ult(V, Ġ) is well-founded” [9].

The central notion of this paper is a strong negation of precipitousness [1]:

Definition. I is nowhere precipitous if I∣X is not precipitous for every X ∈ I+ i.e., ⊨I+ “Ult(V, Ġ) is ill-founded.”

It is useful to characterize nowhere precipitousness in terms of infinite games (see [11, Section 27]). Consider the following game G(I) between two players, Nonempty and Empty [5]. Nonempty and Empty alternately choose Xn ∈ I+ and Yn ∈ I+ respectively so that Xn ⊃ Yn ⊃n+1. After ω moves, Empty wins the game if⋂n<ωXn=⋂n<ωYn = Ø.

See [5, Theorem 2] for a proof of the following characterization.

Proposition. I is nowhere precipitous if and only if Empty has a winning strategy in G(I).

We review some known results about the Ramsey property for partitions of reals, and we present a certain two-person game such that if either player has a winning strategy then a homogeneous set for the partition can be constructed, and conversely. This gives alternative proofs of some of the known results. We then discuss possible uses of the game in obtaining effective versions of Ramsey's theorem and prove a theorem along these lines.

We continue our study of finitary abstract elementary classes, defined in [7]. In this paper, we prove a categoricity transfer theorem for a case of simple finitary AECs. We introduce the concepts of weak κ-categoricity and f-primary models to the framework of ℵ0-stable simple finitary AECs with the extension property, whereby we gain the following theorem: Let () be a simple finitary AEC, weakly categorical in some uncountable κ. Then () is weakly categorical in each λ ≥ min. If the class () is also -tame, weak κ-categoricity is equivalent with κ-categoricity in the usual sense.

We also discuss the relation between finitary AECs and some other non-elementary frameworks and give several examples.

So-called “preservation theorems” relate the (possible) syntactic form of the axioms of a theory to certain closure conditions on its class of models. Such results are well known for the first-order predicate calculus, Lω, ω, and there are various expositions; e.g., Keisler [14], [15]. For the language , the first results were the theorems of Lopez-Escobar on sentences preserved under homomorphic images and of Malitz on formulas preserved under substructures. More recently, Feferman added a result on formulas preserved under (or persistent for) ∈-extensions. Some of these theorems will be considered in subsequent sections. A more thorough treatment may be found in Makkai [17]. The main new preservation result obtained here characterizes the sentences preserved under ω-unions. This notion and the statement of the theorem will be explained shortly.

It is a familiar experience in mathematical research that concepts which are equivalent in a special case diverge in general. In the case at hand, one must expect to consider different possible statements for , which generalize a known result for Lω, ω. Moreover, diverse proofs may yield the same result in the special case, not all of which can be extended to the general case. Again, since the compactness theorem fails for , one cannot expect to extend the arguments from Lω, ω which use this in an essential way.

In this paper we study the automorphism groups of countable arithmetically saturated models of Peano Arithmetic. The automorphism groups of such structures form a rich class of permutation groups. When studying the automorphism group of a model, one is interested to what extent a model is recoverable from its automorphism group. Kossak-Schmerl [12] show that if M is a countable, arithmetically saturated model of Peano Arithmetic, then Aut(M) codes SSy(M). Using that result they prove:

Let M1. M2 be countable arithmetically saturated models of Peano Arithmetic such that Aut(M1) ≅ Aut(M2). Then SSy(M1) = SSy(M2).

We show that if M is a countable arithmetically saturated of Peano Arithmetic, then Aut(M) can recognize if some maximal open subgroup is a stabilizer of a nonstandard element, which is smaller than any nonstandard definable element. That fact is used to show the main theorem:

Let M1, M2be countable arithmetically saturated models of Peano Arithmetic such that Aut(M1) ≅ Aut(M2). Then for every n < ω

Here RT2n is Infinite Ramsey's Theorem stating that every 2-coloring of [ω]n has an infinite homogeneous set. Theorem 0.2 shows that for models of a false arithmetic the converse of Kossak-Schmerl Theorem 0.1 is not true. Using the results of Reverse Mathematics we obtain the following corollary:

There exist four countable arithmetically saturated models of Peano Arithmetic such that they have the same standard system but their automorphism groups are pairwise non-isomorphic.

We study the preservation under projective ccc forcing extensions of the property of L(ℝ) being a Solovay model. We prove that this property is preserved by every strongly- absolutely-ccc forcing extension, and that this is essentially the optimal preservation result, i.e., it does not hold for absolutely-ccc forcing notions. We extend these results to the higher projective classes of ccc posets, and to the class of all projective ccc posets, using definably-Mahlo cardinals. As a consequence we obtain an exact equiconsistency result for generic absoluteness under projective absolutely-ccc forcing notions.

We prove that every diagonally noncomputable function computes a set A which is bi-immune, meaning that neither A nor its complement has an infinite computably enumerable subset.

Kruskal proved that finite trees are well-quasi-ordered by hom(e)omorphic embeddability. Friedman observed that this statement is not provable in predicative analysis. Friedman also proposed (see in [Simpson]) some stronger variants of the Kruskal theorem dealing with finite labeled trees under hom(e)omorphic embeddability with a certain gap-condition, where labels are arbitrary finite ordinals from a fixed initial segment of ω. The corresponding limit statement, expressing that for all initial segments of ω these labeled trees are well-quasi-ordered, is provable in -CA, but not in the analogous theory -CA0 with induction restricted to sets. Schütte and Simpson proved that the one-dimensional case of Friedman's limit statement dealing with finite labeled intervals is not provable in Peano arithmetic. However, Friedman's gap-condition fails for finite trees labeled with transfinite ordinals. In [Gordeev 1] I proposed another gap-condition and proved the resulting one-dimensional modified statements for all (countable) transfinite ordinal-labels. The corresponding universal modified one-dimensional statement UM1 is provable in (in fact, is equivalent to) the familiar theory ATR0 whose proof-theoretic ordinal is Γ0. In [Gordeev 1] I also announced that, in the general case of arbitrarily-branching finite trees labeled with transfinite ordinals, in the proof-theoretic sense the hierarchy of the limit modified statements M<λ (which are denoted by LMλ in the present note) is as strong as the hierarchy of the familiar theories of iterated inductive definitions (more precisely, see [Gordeev 1, Concluding Remark 3]). In this note I present a “positive” proof of the full universal modified statement UM, together with a short proof of the crucial “reverse” results which is based on Okada's interpretation of the well-established ordinal notations of Buchholz corresponding to the theories of iterated inductive definitions. Formally the results are summarized in §5 below.

Higher-order theories of properties, relations, and propositions (PRPs) are known to be essentially incomplete relative to their standard notions of validity. There is, however, a first-order theory of PRPs that results when standard first-order logic is supplemented with an operation of intensional abstraction. It turns out that this first-order theory of PRPs is provably complete with respect to its standard notions of validity. The construction involves the development of a new algebraic semantic method. Unlike most other methods used in contemporary intensional logic, this method does not appeal to possible worlds as a heuristic; the heuristic used is that of PRPs taken as primitive entities. This is important, for even though the possible-worlds approach is useful in treating modal logic, it seems to be of little help in treating the logic for psychological matters. The present approach, by contrast, appears to make a step in the direction of a satisfactory treatment of both modal and intentional logic. For, by taking PRPs as primitive entities, we remain free to tailor the statement of their identity conditions so that it agrees with the logical data—modal, psychological, etc. In this way, the present approach suggests a strategy for developing a comprehensive treatment of intensional logic.

In [1] and [2] I explore this prospect philosophically. The purpose of the present paper is to lay out the technical details of the approach and to present the completeness results.

By ℬ2 we denote the σ-ideal of all subsets A of the Cantor set {0, 1}ω such that for every infinite subset T of ω the restriction A∣{0, 1}T is a proper subset of {0, 1}T. In this paper we investigate set theoretical properties of this and similar ideals.

Let T be a complete O-minimal theory in a language L. We first give an elementary proof of the result (due to Marker and Steinhorn) that all types over Dedekind complete models of T are definable. Let L* be L together with a unary predicate P. Let T* be the L*-theory of all pairs (N, M), where M is a Dedekind complete model of T and N is an ⅼMⅼ+-saturated elementary extension of N (and M is the interpretation of P). Using the definability of types result, we show that T* is complete and we give a simple set of axioms for T*. We also show that for every L*-formula ϕ(x) there is an L-formula ψ(x) such that T* ⊢ (∀x)(P(x) → (ϕ(x) ↔ ψ(x)). This yields the following result:

Let M be a Dedekind complete model of T. Let ϕ(x, y) be an L-formula where l(y) – k. Let X = {X ⊂ Mk: for some a in an elementary extension N of M, X = ϕ(a, y)N ∩ Mk}. Then there is a formula ψ(y, z) of L such that X = {ψ(y, b)M: b in M}.

Ramsey's Theorem states that if P is a partition of [ω]k into finitely many partition classes, then there exists an infinite set of natural numbers which is homogeneous for P. We consider the degrees of unsolvability and arithmetical definability properties of infinite homogeneous sets for recursive partitions. We give Jockusch's proof of Seetapun's recent theorem that for all recursive partitions of [ω]2 into finitely many pieces, there exists an infinite homogeneous set A such that ∅′ ≰TA. Two technical extensions of this result are given, establishing arithmetical bounds for such a set A. Applications to reverse mathematics and introreducible sets are discussed.

This is the second part of a study on model theory of modules begun in [RO]. Throughout, I refer to that paper as “Part I”. I observed there a coincidence between some algebraic and logical points of view in the theory of modules, which led to a convenient representation of p.p. definable sets in flat modules (§1); it becomes especially nice in the case of regular rings (cf. Remark 7 in Part I). Using this in the present paper I obtain a simplified criterion for total transcendence and superstability in the case of flat modules. This enables me to give, besides partial results for arbitrary flat modules (§2), a complete description of the stability classes for modules over regular rings (§3). Particularly, it turns out that a totally transcendental module over a regular ring can be regarded as a module over a semisimple ring (cf. Remark 10 in Part I). This is the crucial observation for the examination of categoricity here: By Morley's theorem, a countable ℵ1-categorical theory is totally transcendental. Consequently, an ℵ1-categorical module over a countable regular ring can be regarded as a module over a semisimple ring. That is why I separately treat categoricity of modules over semisimple rings (§4), even without any assumption on the power of the ring. As a consequence I obtain a complete description of ℵ1-categorical modules over countable regular rings (§5). In the investigation presented here the main tool is the technique of idempotents avoiding the commutativity assumption made in the corresponding results of Garavaglia [GA 1, pp. 86–88], who used maximal ideals for that purpose. At the end of the present paper I show how to derive these latter results in our context.

On the way I simplify the known criterion for elementary equivalence for modules over regular rings (§3), which simplifies again in case of semisimple rings (§4). Such a criterion is needed in order to construct Vaughtian pairs (in the categoricity consideration) and it turns out to be useful also for another purpose treated in the third part of this series of papers.

A family A ⊆ (ω) is called countably splitting if for every countable F ⊆ [ω]ω, some element of A splits every member of F. We define a notion of a splitting tree, by means of which we prove that every analytic countably splitting family contains a closed countably splitting family. An application of this notion solves a problem of Blass. On the other hand we show that there exists an Fσ splitting family that does not contain a closed splitting family.

We improve results of Marczewski, Frankiewicz, Brown, and others comparing the ω-ideals of measure zero, meager, Marczewski measure zero, and completely Ramsey null sets; in particular, we remove CH from the hypothesis of many of Brown's constructions of sets lying in some of these ideals but not in others. We improve upon work of Marczewski by constructing, without CH, a nonmeasurable Marczewski measure zero set lacking the property of Baire. We extend our analysis of ω-ideals to include the completely Ramsey null sets relative to a Ramsey ultrafilter and obtain all 32 possible examples of sets in some ideals and not others, some under the assumption of MA, but most in ZFC alone. We also improve upon the known constructions of a Marczewski measure zero set which is not Ramsey by using a set theoretic hypothesis which is weaker than those used by other authors. We give several consistency proofs: one concerning the relative sizes of the covering numbers for the meager sets and the completely Ramsey null sets; another concerning the size of non; and a third concerning the size of add. We also study those classes of perfect sets which are bases for the class of always first category sets.

It is shown that the constructive four-valued logic N4 can be faithfully embedded into the modal logic S4. This embedding is used to obtain complete, cut-free display sequent calculi for N4 and C4, the modal logic of consistency over N4. C4 is a natural monotonic base system for semantics-based non-monotonic reasoning.

We show that the Π4-theory of the partial order of recursively enumerable weak truth-table degrees is undecidable, and give a new proof of the similar fact for r.e. T-degrees. This is accomplished by introducing a new coding scheme which consists in defining the class of finite bipartite graphs with parameters.

A method of iteration of Prikry type forcing notions as well as a forcing for adding clubs is presented. It is applied to construct a model with a measurable cardinal containing a club of former regulars, starting with ο(κ) = κ + 1. On the other hand, it is shown that the strength of above is at least

ο(κ) = κ.

We study ultrafilters on ω2 produced by forcing with the quotient of ${\cal P}$(ω2) by the Fubini square of the Fréchet filter on ω. We show that such an ultrafilter is a weak P-point but not a P-point and that the only nonprincipal ultrafilters strictly below it in the Rudin–Keisler order are a single isomorphism class of selective ultrafilters. We further show that it enjoys the strongest square-bracket partition relations that are possible for a non-P-point. We show that it is not basically generated but that it shares with basically generated ultrafilters the property of not being at the top of the Tukey ordering. In fact, it is not Tukey-above [ω1]<ω, and it has only continuum many ultrafilters Tukey-below it. A tool in our proofs is the analysis of similar (but not the same) properties for ultrafilters obtained as the sum, over a selective ultrafilter, of nonisomorphic selective ultrafilters.