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.

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.

Let Z2, Z3, and Z4 denote 2nd, 3rd, and 4th order arithmetic, respectively. We let Harrington’s Principle, HP, denote the statement that there is a real x such that every x-admissible ordinal is a cardinal in L. The known proofs of Harrington’s theorem “$Det\left( {{\rm{\Sigma }}_1^1} \right)$ implies 0♯ exists” are done in two steps: first show that $Det\left( {{\rm{\Sigma }}_1^1} \right)$ implies HP, and then show that HP implies 0♯ exists. The first step is provable in Z2. In this paper we show that Z2 + HP is equiconsistent with ZFC and that Z3 + HP is equiconsistent with ZFC + there exists a remarkable cardinal. As a corollary, Z3 + HP does not imply 0♯ exists, whereas Z4 + HP does. We also study strengthenings of Harrington’s Principle over 2nd and 3rd order arithmetic.

We study some generalized notions of cohesiveness which arise naturally in connection with effective versions of Ramsey's Theorem. An infinite set A of natural numbers is n-cohesive (respectively, n-r-cohesive) if A is almost homogeneous for every computably enumerable (respectively, computable) 2-coloring of the n-element sets of natural numbers. (Thus the 1-cohesive and 1-r-cohesive sets coincide with the cohesive and r-cohesive sets, respectively.) We consider the degrees of unsolvability and arithmetical definability levels of n-cohesive and n-r-cohesive sets. For example, we show that for all n ≥ 2, there exists a n-cohesive set. We improve this result for n = 2 by showing that there is a 2-cohesive set. We show that the n-cohesive and n-r-cohesive degrees together form a linear, non-collapsing hierarchy of degrees for n ≥ 2. In addition, for n ≥ 2 we characterize the jumps of n-cohesive degrees as exactly the degrees ≥ 0(n+1) and also characterize the jumps of the n-r-cohesive degrees.

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.

Using a dictionary translating a variety of classical and modern covering properties into combinatorial properties of continuous images, we get a simple way to understand the interrelations between these properties in ZFC and in the realm of the trichotomy axiom for upward closed families of sets of natural numbers. While it is now known that the answer to the Hurewicz 1927 problem is positive, it is shown here that semifilter trichotomy implies a negative answer to a slightly stronger form of this problem.

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.

The field K((G)) of generalized power series with coefficients in the field K of characteristic 0 and exponents in the ordered additive abelian group G plays an important role in the study of real closed fields. Conway and Gonshor (see [2, 4]) considered the problem of existence of non-standard irreducible (respectively prime) elements in the huge “ring” of omnific integers, which is indeed equivalent to the existence of irreducible (respectively prime) elements in the ring K((G≤0)) of series with non-positive exponents. Berarducci (see [1]) proved that K((G≤0)) does have irreducible elements, but it remained open whether the irreducibles are prime i.e., generate a prime ideal. In this paper we prove that K((G≤0)) does have prime elements if G = (ℝ, +) is the additive group of the reals, or more generally if G contains a maximal proper convex subgroup.

In this article an application of extended fields of real numbers to the proof of theorems in the theory of cooperative games will be presented. The proofs set forth below involve the use of A. Robinson's theory of nonstandard analysis and are metamathematical in character. Alternative proofs utilizing ordinary topological methods can in fact be carried out quite briefly. However, attempts to apply nonstandard analysis to game theory are relatively novel. For this reason these results may be of interest not only insofar as they present new information on the theory of the kernel of a cooperative game, but also in that they serve to demonstrate the possibility of effectively exploiting nonstandard analysis as a tool for future investigation in this area. It may well turn out that nonstandard analysis could serve as the most natural vehicle through which concepts defined for games with a finite number of players might be extended to games with a continuum of players.

1. The problem. The aim of the following considerations is to introduce a new type of non-Aristotelian logic by generalizing the truth-table methods so far employed for establishing non-Aristotelian sentential calculi. We shall expound the intended generalization by applying it to the particular set of pluri-valued systems introduced by J. Łukasiewicz. One will remark that the points of view illustrated by this example may serve to generalize quite analogously any other plurivalued systems, such as those originated by E. L. Post, by H. Reichenbach, and by others.

2. J. Łukasiewicz's plurivalued systems of sentential logic. First of all, we consider briefly the structure of the Łukasiewicz systems themselves.

As to the symbolic notation in which to represent those systems, we make the following agreements: For representing the expressions of the (two- or plurivalued) calculus of sentences, we make use of the Principia mathematica symbolism; however, we employ brackets instead of dots. We call the small italic letters “p”, “q”, “r”, … sentential variables or elementary sentences, and employ the term “sentence” as a general designation of both elementary sentences and the composites made up of elementary sentences and connective symbols (“~”, “ν” “.”, “⊃” “≡”).

Now, the different possible sentences (or, properly speaking, the different possible shapes of sentences, such as “p”, “p∨q”, “~p.(q∨ r)”, etc.) are the objects to which truth-values are ascribed; and just as in every other case one wants a designation for an object in order to be able to speak of it, we want now a system of designations for the sentences with which we are going to deal in our truth-table considerations.

It is shown to be consistent that the reals are covered by ℵ1, meagre sets yet there is a Baire class 1 function which cannot be covered by fewer than ℵ2, continuous functions. A new cardinal invariant is introduced which corresponds to the least number of continuous functions required to cover a given function. This is characterized combinatorially. A forcing notion similar to, but not equivalent to, superperfect forcing is introduced.

We formulate and prove (in ZFC) a strong coloring theorem which holds at successors of singular cardinals, and use it to answer several questions concerning Shelah's principle Pr1 (μ+, μ+, μ+, cf (μ)) for singular μ.

Throughout, M will denote a transitive model of ZFC. Using the terms “random” and “generic” in the sense of [1], one may ask whether there can exist real numbers x and y such that x is generic over M[y] and y is random over M[x]. We shall see below by an elementary argument that this is not possible, and so, in a crude sense at least, random and generic extensions do not commute. This does not however rule out the possibility of a weaker commutativity. Let B be the complete Boolean algebra (in M) for adjoining a random real followed by a generic real and C be the complete Boolean algebra for adjoining a generic real followed by a random real. Then it still might be the case that B and C are isomorphic. This also fails, though, and we shall prove this by establishing the following combinatorial properties of MB and MC:

but

In addition this will show that C cannot be embedded as a complete subalgebra of B.

The property satisfied by B is reminiscent of calibre ℵ1 [2]. B would have calibre if we could replace “infinite” by “uncountable”, and this occurs if Martin's Axiom holds in M. To obtain the nonisomorphism of B and C in general necessitated looking at the weaker property.

Cartesian closed categories (CCCs) have played and continue to play an important role in the study of the semantics of programming languages. An axiomatization of the isomorphisms which hold in all Cartesian closed categories discovered independently by Soloviev and Bruce, Di Cosmo and Longo leads to seven equalities. We show that the unification problem for this theory is undecidable, thus settling an open question. We also show that an important subcase, namely unification modulo the linear isomorphisms, is NP-complete. Furthermore, the problem of matching in CCCs is NP-complete when the subject term is irreducible.

CCC-matching and unification form the basis for an elegant and practical solution to the problem of retrieving functions from a library indexed by types investigated by Rittri. It also has potential applications to the problem of polymorphic type inference and polymorphic higher-order unification, which in turn is relevant to theorem proving and logic programming.

We use a fundamental theorem of Vaught, called the covering theorem in [V] (cf. theorem 0.1 below) as well as a generalization of it (cf. Theorem 0.1* below) to derive several known and a few new results related to the logic . Among others, we prove that if every countable model in a PCφ1ω class has only countably many automorphisms, then the class has either ≤ℵ0 or exactly nonisomorphic countable members (cf. Theorem 4.3*) and that the class of countable saturated structures of a sufficiently large countable similarity type is not among countable structures (cf. Theorem 5.2). We also give a simple proof of the Lachlan-Sacks theorem on bounds of Morley ranks (§7).

We study classes of ultrafilters on ω defined by a natural property of the Loeb measure in the Nonstandard Universe corresponding to the ultrafilter. This class, the Property M ultrafilters, is shown to contain all ultrafilters built up by taking iterated products over collections of pairwise nonisomorphic selective ultrafilters. Results on Property M ultrafilters are applied to the construction of extensions of probability measures, and to the study of measurable reductions between ultrafilters.

In analogy to the definition of the halting problem K, an r.e. set A is called a diagonal iff there is a computable numbering ψ of the class of all partial recursive functions such that A = {i ∈ ω: ψi(i)↓} (in that case we say that A is the diagonal of ψ). This notion has been introduced in [10]. It captures all r.e. sets that can be constructed by diagonalization.

It was shown that any nonrecursive r.e. T-degree contains a diagonal, that for any diagonal A there is an r.e. nonrecursive nondiagonal B ≤TA, and that there are r.e. degrees a such that any r.e. set from a is a diagonal.

In §2 of the present paper we show that the property “A is a diagonal” is elementary lattice theoretic (e.l.t.). This result complements and generalizes previous results of Harrington and Lachlan, respectively. Harrington (see [16, XV. 1]) proved that the property of being the diagonal of some Gödelnumbering (i.e., of being creative) is e.l.t., and Lachlan [12] proved that the property of being a simple diagonal (i.e., of being simple and not hh-simple [10]) is e.l.t.

In §§3 and 4 we study the position of diagonals and nondiagonals inside the lattice of r.e. sets. We concentrate on an important class of nondiagonals, generalizing the maximal and hemimaximal sets: the -maximal sets. Using the results from [6] we are able to classify the -maximal sets that can be obtained as halfs of splittings of hh-simple sets.

Given a finitely generated group Γ we study the space Isom(Γ, ) of all actions of Γ by isometries of the rational Urysohn metric space , where Isom (Γ, ) is equipped with the topology it inherits seen as a closed subset of Isom . When Γ is the free group on n generators this space is just Isom , but is in general significantly more complicated. We prove that when Γ is finitely generated Abelian there is a generic point in Isom(Γ, ), i.e., there is a comeagre set of mutually conjugate isometric actions of Γ on .