It has been proven by Blass [1973] that any two P-points which have a P-point as a common upper bound in the Rudin-Keisler (RK) ordering necessarily have a common lower bound (necessarily a P-point). Hence two nonisomorphic Ramsey ultrafilters have neither a common lower nor a common upper bound which is a P-point. So in a model of CH for example (or MA, P(c),…), the RK ordering restricted to P-points is neither upward nor downward directed, since it is well known that nonisomorphic Ramsey ultrafilters exist in such models. On the other hand, we will see that in the model for “near coherence of filters” (NCF) produced by Blass and Shelah [1985], the RK ordering of P-points is upward, hence downward directed. This shows that the question of directedness of the RK ordering of P-points, upward or downward, cannot be decided in ZFC.

There is a related question, asked by Blass in [1973], whether two P-points which have a common lower bound necessarily have a common upper bound which is a P-point. Our main result establishes the independence of this statement relative to ZFC. Its consistency will follow as soon as we show that the RK ordering of P-points is upward directed in the NCF model mentioned above, which we do in §2. But its independence will require a new construction, and will be given in §3.

We prove the existence of p-ideals that are nonmeagre subsets of (ω) under various set-theoretic assumptions.

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.

Before the remarkable theorem of Martin and Steel [6] showing that the existence of a supercompact cardinal κ implies L[R] ⊨ ZF + AD + DC, and the later theorem of Woodin [9] showing that Con(ZFC + There exists an ω sequence of Woodin cardinals) ⇔ Con(ZF + AD + DC), much set-theoretic research was focused upon showing that the consistency of fragments of AD + DC followed from more “reasonable” hypotheses such as versions of supercompactness. A good example of this is provided by the results of [1], in which the following theorems are proven.

We characterize, in terms of determinacy, the existence of 0## as well as the existence of each of the following: 0###, 0####, 0#####,…. For κ Є ω, we define two classes of sets, and , which lie strictly between and . We also define 01# as# and in general, 0(k+1)# as (0k#)#. We then show that the existence of 0(k + 1)# is equivalent to the determinacy of as well as the determinacy of .

Higher Suslin trees have become a tool in some forcing constructions in set theory (see, for example, [D1] and [D2]). Most of the constructions using ω1 Suslin trees can be extended to κ+ Suslin trees for any regular cardinal κ. Some of these are given in §1.

In many such constructions, sequences of Suslin trees are used. In §II we show, in various ways, that the generalization to sequences, even ω-sequences, of κ+ Suslin trees cannot be done.

In these constructions the Suslin trees are used as forcing poset (the forcing adds a branch in the tree). There is another way to kill a Suslin tree, namely by adding a big antichain. Some results on this forcing are given in §III.

Our notation is standard. If T is a tree and x ∈ T, then ∣x∣ is the height of x in T. We define Tα (or T(α)) = {x ∈ T: ∣x∣ = α} and T∣α = {x ∈ T: ∣x∣ < α}.

If p and q are forcing conditions, p ≤ q means that p has more information than q.

If (Tα: α ∈ I) is a sequence of trees, Π Tα will always mean the set of (xα: α ∈ I) such that xα ∈ Tα and ∣xα∣ = ∣xβ∣ for α, β ∈ I.

For functions b, T,… we denote by b ∣α, T∣ α,… their restriction to α.

If x is a sequence of ordinals and α is an ordinal, x∧α is the sequence obtained by concatenating α at the end of x.

We provide partial answers to the following problem: Is the class of Borel linear orders well-quasi-ordered under embeddability? We show that it is indeed the case for those Borel orders which are embeddable in Rω, with the lexicographic ordering. For Borel orders embeddable in R2, our proof works in ZFC, but it uses projective determinacy for Borel orders embeddable in some Rn, n < ω, and hyperprojective determinacy for the general case.

The algebraic systems of combinatory spaces [3] and operative spaces [1] have been designed to provide appropriate settings for the development of abstract recursion theory. As shown in [1, Chapter 27], these systems are closely related; namely, every combinatory space has a companion operative space with a storing operation St such that Skordev recursiveness in the former equals st-recursiveness in the latter. The problem of characterization of those operative spaces which have companion combinatory spaces was solved in [2] by introducing a class of operative spaces called Skordev spaces and showing that these give an alternative axiomatics of Skordev's combinatory spaces.

In this paper we study operative spaces with two special constants S1 and S2 which, in computer science terms, axiomatize the availability of stacking facility. We establish that any such space has a Skordev subspace and, moreover, all Skordev spaces can be obtained in this way. The construction ensures that st-recursiveness in the Skordev space concerned is equivalent to recursiveness in S1 and S2 in the wider operative space. Therefore, it quite unexpectedly turns out that Skordev recursiveness can be expressed via relative recursiveness in operative spaces in a straightforward manner, without employing higher spaces as done in [1].

Let β be an arbitrary limit ordinal. A β-r.e. set is l-finite iff all its β-r.e. subsets are β-recursive. The l-finite sets correspond to the ideal of finite sets in the lattice of r.e. sets. We give a characterization of l-finite sets in terms of their ordertype: a β-r.e. set is l-finite iff it has ordertype less than β*, the Σ1, projectum of β).

In this note we prove that the abstract property “weakly subdirectly indecomposable” does not characterize the class IWsα of weak cylindric set algebras. However, we give another (similar) abstract property characterizing IWsα. The original property does characterize the directed unions of members of IWsα iff α is countable. Free algebras will be shown to satisfy the original property.

A topological classification scheme consists of two ingredients: (1) an abstract class of topological spaces; and (2) a “taxonomy”, i.e. a list of first order sentences, together with a way of assigning an abstract class of spaces to each sentence of the list so that logically equivalent sentences are assigned the same class. is then endowed with an equivalence relation, two spaces belonging to the same equivalence class if and only if they lie in the same classes prescribed by the taxonomy. A space X in is characterized within the classification scheme if whenever Y ∊ and Y is equivalent to X, then Y is homeomorphic to X. As prime example, the closed set taxonomy assigns to each sentence in the first order language of bounded lattices the class of topological spaces whose lattices of closed sets satisfy that sentence. It turns out that every compact two-complex is characterized via this taxonomy in the class of metrizable spaces, but that no infinite discrete space is so characterized. We investigate various natural classification schemes, compare them, and look into the question of which spaces can and cannot be characterized within them.

A remarkable result of Henson and Ross [HR] states that if a function whose graph is Souslin in the product of two hyperfinite sets in an ℵ1 saturated nonstandard universe possesses a certain nice property (capacity) then there exists an internal subfunction of the given one possessing the same property. In particular, they prove that every 1-1 Souslin function preserves any internal counting measure, and show that every two internal sets A and B with ∣A∣/∣B∣ ≈ 1 are Borel bijective. As a supplement to the last-mentioned result of Henson and Ross, Keisler, Kunen, Miller and Leth showed [KKML] that two internal sets A and B are bijective by a countably determined bijection if and only if ∣A∣/∣B∣ is finite and not infinitesimal.

In this paper we first show that injective Borel functions map Borel sets into Borel sets, a fact well known in classical descriptive set theory. Then, we extend the result of Henson and Ross concerning the Borel bijectivity of internal sets whose quotient of cardinalities is infinitely closed to 1. We prove that two Borel sets, to which we may assign a counting measure not equal to 0 or ∞, are Borel bijective if and only if they have the same counting measure ≠0, ∞. This, together with the similar characterization for Souslin and measurable countably determined sets, extends the above-mentioned results from [HR] and [KKML].

We prove that no logic (i.e. consequence operation) determined by any class of orthomodular lattices admits the deduction theorem (Theorem 2.7). We extend those results to some broader class of logics determined by ortholattices (Corollary 2.6).

Let T be a complete countable first order theory and λ an uncountable cardinal. Theorem 1. If T is not superstable, T has 2λ resplendent models of power λ. Theorem 2. If T is strictly superstable, then T has at least min(2λ, ℶ2) resplendent models of power λ. Theorem 3. If T is not superstable or is small and strictly superstable, then every resplendent homogeneous model of T is saturated. Theorem 4 (with Knight). For each μ ∈ ω ∪ {ω, 2ω} there is a recursive theory in a finite language which has μ resplendent models of power κ for every infinite κ.

We introduce the notion of “semi-r.e.” for subsets of ω, a generalization of “semirecursive” and of “r.e.”, and the notion of “weakly semirecursive”, a generalization of “semi-r.e.”. We show that A is weakly semirecursive iff, for any n numbers x1, …,xn, knowing how many of these numbers belong to A is equivalent to knowing which of these numbers belong to A. It is shown that there exist weakly semirecursive sets that are neither semi-r.e. nor co-semi-r.e. On the other hand, we exhibit nonzero Turing degrees in which every weakly semirecursive set is semirecursive. We characterize the notion “A is weakly semirecursive and recursive in K” in terms of recursive approximations to A. We also show that if a finite Boolean combination of r.e. sets is semirecursive then it must be r.e. or co-r.e. Several open questions are raised.

Several definitions of “compact” for topological spaces have appeared in the literature (see [5]). We will consider the following:

Definition. A topological space X is

1. Compact(1) if every open cover of X has a finite subcover.

2. Compact(2) if every infinite subset E of X has a complete accumulation point (i.e., a point x0 ∈ X such that for every neighborhood U of x0, |E ∩ U| = |E|).

3. Compact(3) if there is a subbase S for the topology on X such that every cover of X by members of S has a finite subcover.

4. Compact(4) if each nest of closed, nonempty sets has a nonempty intersection.

5. Compact(5) if every family of closed sets in X which has the finite intersection property (every finite subfamily has a nonempty intersection) has a nonempty intersection.

6. Compact(6) if each net in X has a cluster point.

7. Compact(7) if each net in X has a convergent subnet.

This work was motivated primarily by consideration of various proofs that the Tychonoff theorem, T (“the product of compact topological spaces is compact”) is equivalent to the Axiom of Choice, AC. Tychonoff's original proof that AC implies T used Definition 2 [13]. Other proofs have used Definitions 3 and 5; see [5]. The proof by Kelley that T implies AC uses Definition 5 [6].

A computer handles λ-terms more easily if these are translated into combinatory terms. This translation process is called bracket abstraction. The simplest abstraction algorithm—the (fab) algorithm of Curry (see Curry and Feys [6])—is lengthy to implement and produces combinatory terms that increase rapidly in length as the number of variables to be abstracted increases.

There are several ways in which these problems can be alleviated:

(1) A change in order of the clauses in the algorithm so that (f) is performed as a last resort.

(2) The use of an extra clause (c), appropriate to βη reduction.

(3) The introduction of a finite number of extra combinators.

The original 1924 form of bracket abstraction of Schönfinkel [17], which in fact predates λ-calculus, uses all three of these techniques; all are also mentioned in Curry and Feys [6].

A technique employed by many computing scientists (Turner [20], Peyton Jones [16], Oberhauser [15]) is to use the (fab) algorithm followed by certain “optimizations” or simplifications involving extra combinators and sometimes special cases of (c).

Another is either to allow a fixed infinite set of (super-) combinators (Abdali [1], Kennaway and Sleep [10], Krishnamurthy [12], Tonino [19]) or to allow new combinators to be defined one by one during the abstraction process (Hughes [7] and [8]).

A final method encodes the variables to be abstracted as an n-tuple—this requires only a finite number of combinators (Curien [5], Statman [18]).

In what follows, a coset is a subset of a group G of the form aH, where H is a subgroup of G; H can be recovered from the coset C: it is the only subgroup which is obtained from C by a left translation; we note in passing that these cosets, that we write systematically with the group to the right, are also of the form Ka, since aH = aHa−1a. A classical combinatorial lemma involving cosets appears in Neumann [1952]: If the coset C = aH is the union of the finite family of cosets C1 = a1H1,…,Cn = anHn, then it is the union of those Ci whose corresponding Hi has finite index in H.

In a structure where a group G is defined, Boolean combinations of cosets modulo its definable subgroups form a family of definable sets (by definable, we mean “definable with parameters”). The situation when any definable set is of that kind has been characterized model-theoretically in Hrushovski and Pillay [1987]: A group G is one-based if and only if, for each n, every definable subset of the cartesian power Gn is a(finite!) Boolean combination of cosets modulo definable subgroups. One side is given by a beautiful lemma of Pillay, stating that, in a one-based group which is saturated enough, every type is a right translate of the generic of its left stabilizer.

Any finite support iteration of posets with precalibre ℵ1 which has the length of cofinahty greater than ω1 yields a model for the dual Borel conjecture in which the real line is covered by ℵ1 strong measure zero sets.