A classic result of Baumgartner-Harrington-Kleinberg [1] implies that assuming CH a stationary subset of ω1 has a CUB subset in a cardinal-perserving generic extension of V, via a forcing of cardinality ω1. Therefore, assuming that ω2L is countable: {X ∈ L ∣ X ⊆ ω1L and X has a CUB subset in a cardinal-preserving extension of L} is constructive, as it equals the set of constructible subsets of ω1L which in L are stationary. Is there a similar such result for subsets of ω2L? Building on work of M. Stanley [9], we show that there is not. We shall also consider a number of related problems, examining the extent to which they are “solvable” in the above sense, as well as denning a notion of reduction between them.

In the constructive theory of uniform spaces there occurs a technique of proof in which the application of a weak form of the law of excluded middle is circumvented by purely analytic means. The essence of this proof–technique is extracted and then applied in several different situations.

We reprove Gitik's theorem that if the GCH holds and o(κ) = κ + 1 then there is a generic extension in which κ is still measurable and there is a closed unbounded subset C of κ such that every ν ∈ C is inaccessible in the ground model.

Unlike the forcing used by Gitik, the iterated forcing ℛλ+1 used in this paper has the property that if λ is a cardinal less then κ then ℛλ+1 can be factored in V as ℛκ+1 = ℛλ+1 × ℛλ+1,κ where ∣ℛλ+1∣ ≤ λ+ and ℛλ+1,κ does not add any new subsets of λ.

This paper developed from Shelah's proof of a zero-one law for the complexity class “choiceless polynomial time,” defined by Shelah and the authors. We present a detailed proof of Shelah's result for graphs, and describe the extent of its generalizability to other sorts of structures. The extension axioms, which form the basis for earlier zero-one laws (for first-order logic, fixed-point logic, and finite-variable infinitary logic) are inadequate in the case of choiceless polynomial time; they must be replaced by what we call the strong extension axioms. We present an extensive discussion of these axioms and their role both in the zero-one law and in general.

The theme of this paper is the generalization of theorems about partitions of the sets of points and lines of finite-dimensional Euclidean spaces ℝd to vector spaces over ℝ of arbitrary dimension and, more generally still, to arbitrary vector spaces over other fields so long as these fields are not too big. These theorems have their origins in the following striking theorem of Sierpiński [12] which appeared a half century ago.

Sierpiński's Theorem. The Continuum Hypothesis is equivalent to: There is a partition {X, Y, Z} of ℝ3such that if ℓ is a line parallel to the x-axis [respectively: y-axis, z-axis] then X ∩ ℓ [respectively: Y ∩ ℓ, Z ∩ ℓ] is finite.

The history of this theorem and some of its subsequent developments are discussed in the very interesting article by Simms [13]. Sierpiński's Theorem was generalized by Kuratowski [9] to partitions of ℝn+2 into n + 2 sets obtaining an equivalence with . The geometric character that Sierpiński's Theorem and its generalization by Kuratowski appear to have is bogus, since the lines parallel to coordinate axes are essentially combinatorial, rather than geometric, objects. The following version of Kuratowski's theorem emphasizes its combinatorial character.

Kuratowski's Theorem. Let n < ω and A be any set. Then ∣A∣ ≤ ℵnif and only if there is a partition P: An+2 → n + 2 such that if i ≤ n + 1 and ℓ is a line parallel to the i-th coordinate axis, then {x ∈ ℓ: P(x) = i} is finite.

We prove linear and polynomial growth properties of sets and functions that are existentially definable in the ordered group of integers with divisibility. We determine the laws of addition with order and divisibility.

For n < ω, expand the structure (n, S, I, F) (with S the successor relation, I, F as the initial and final element) by forming graphs with edge probability n−α for irrational α, with 0 < α < 1. The sentences in the expanded language, which have limit probability 1, form a complete and stable theory.

Dual combinators emerge from the aim of assigning formulas containing ← as types to combinators. This paper investigates formally some of the properties of combinatory systems that include both combinators and dual combinators. Although the addition of dual combinators to a combinatory system does not affect the unique decomposition of terms, it turns out that some terms might be redexes in two ways (with a combinator as its head, and with a dual combinator as its head). We prove a general theorem stating that no dual combinatory system possesses the Church-Rosser property. Although the lack of confluence might be problematic in some cases, it is not a problem per se. In particular, we show that no damage is inflicted upon the structurally free logics, the system in which dual combinators first appeared.

In this paper, following an idea of Christophe Chalons, I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence φ holding in some forcing extension Vℙ and all subsequent extensions Vℙ*ℚ holds already in V. It follows, in fact, that such sentences must also hold in all forcing extensions of V. In modal terms, therefore, the Maximality Principle is expressed by the scheme (◊ □ φ) ⇒ □ φ, and is equivalent to the modal theory S5. In this article, I prove that the Maximality Principle is relatively consistent with ZFC. A boldface version of the Maximality Principle, obtained by allowing real parameters to appear in φ, is equiconsistent with the scheme asserting that Vδ ≺ V for an inaccessible cardinal δ, which in turn is equiconsistent with the scheme asserting that ORD is Mahlo. The strongest principle along these lines is □ , which asserts that holds in V and all forcing extensions. From this, it follows that 0# exists, that x# exists for every set x, that projective truth is invariant by forcing, that Woodin cardinals are consistent and much more. Many open questions remain.

We prove a cell decomposition theorem for Presburger sets and introduce a dimension theory for Z-groups with the Presburger structure. Using the cell decomposition theorem we obtain a full classification of Presburger sets up to definable bijection. We also exhibit a tight connection between the definable sets in an arbitrary p-minimal field and Presburger sets in its value group. We give a negative result about expansions of Presburger structures and prove uniform elimination of imaginaries for Presburger structures within the Presburger language.

It is proved in this paper that the positive abstraction scheme is consistent with extensionality only if one drops equality out of the language. The theory obtained is then compared with GPK, a well-known set theory based on an extended positive comprehension scheme.

We describe the Ax-Kochen definable subsets of the value group of a Hensel field and apply our results to a problem on identifying invariant factors in Hecke algebras.

We strengthen a theorem of Gitik and Shelah [6] by showing that if κ is either weakly inaccessible or the successor of a singular cardinal and S is a stationary subset of κ such that NSκ↾S is saturated then κ ∖ S is fat. Using this theorem we derive some results about the existence of fat stationary sets. We then strengthen some results due to Baumgartner and Taylor [2], showing in particular that if I is a λ+++-saturated normal ideal on Pκλ then the conditions of being λ+-preserving, weakly presaturated, and presaturated are equivalent for I.

We show that the ‘tail’ of a doubly homogeneous chain of countable cofinality can be recognized in the quotient of its automorphism group by the subgroup consisting of those elements whose support is bounded above. This extends the authors' earlier result establishing this for the rationals and reals. We deduce that any group is isomorphic to the outer automorphism group of some simple lattice-ordered group.

Kaneko-Suzuki developed epistemic logics of shallow depths with multiple players for investigations of game theoretical problems. By shallow depth, we mean that nested occurrences of belief operators of players in formulae are restricted, typically to be of finite depths, by a given epistemic structure. In this paper, we develop various methods of surgical operations (cut and paste) of epistemic world models. An example is a bouquet-making, i.e., tying several models into a bouquet. Another example is to engraft a model to some branches of another model. By these methods, we obtain various meta-theorems on semantics and syntax on epistemic logics. To illustrate possible uses of our meta-theorems, we present one game theoretical theorem, which is also a meta-theorem in the sense of logic.

Given two incomparable c.e. Turing degrees a and b, we show that there exists a c.e. degree c such that c = (a ∪ c) ∩ (b ∪ c), a ∪ c ∣ b ∪ c, and c < a ∪ b.

Given well ordered countable sets of the form Λϕ, we consider Borel mappings from Λϕω with countable image inside the ordinals. The ordinals and Λϕω are respectively equipped with the discrete topology and the product of the discrete topology on Λϕ. The Steel well-ordering on such mappings is denned by ϕ ≤ ψ iff there exists a continuous function f such that ϕ(x) ≤ ψof(x) holds for any x ϵ Λϕω. It induces a hierarchy of mappings which we give a complete description of. We provide, for each ordinal α, a mapping whose rank is precisely α in this hierarchy and we also compute the height of the hierarchy restricted to mappings with image bounded by α. These mappings being viewed as partitions of the reals, there is, in most cases, a unique distinguished element of the partition. We analyze the relation between its topological complexity and the rank of the mapping in this hierarchy.

Dividing independence for ultraimaginaries is neither symmetric nor transitive. Moreover, any notion of independence satisfying certain axioms (weaker than those for independence in a simple theory) and denned for all ultraimaginary sorts, is necessarily trivial.

We characterize the classifiability of a countable first-order theory T in terms of the solvability (in the sense of [2]) of the potential-isomorphism problem for models of T.

Cholak, Goncharov, Khoussainov, and Shore [1] showed that for each k > 0 there is a computably categorical structure whose expansion by a constant has computable dimension k. We show that the same is true with k replaced by ω. Our proof uses a version of Goncharov's method of left and right operations.