Shelah-Woodin [10] investigate the possibility of violating instances of GCH through the addition of a single real. In particular they show that it is possible to obtain a failure of CH by adding a single real to a model of GCH, preserving cofinalities. In this article we strengthen their result by showing that it is possible to violate GCH at all infinite cardinals by adding a single real to a model of GCH. Our assumption is the existence of an H(κ+3)-strong cardinal; by work of Gitik and Mitchell [6] it is known that more than an H(κ++)-strong cardinal is required.

It is shown that the power set of κ ordered by the subset relation modulo various versions of the non-stationary ideal can be embedded into the partial order of Borel equivalence relations on 2κ under Borel reducibility. Here κ is an uncountable regular cardinal with κ<κ = κ.

Recently, Gitik, Kanovei and the first author proved that for a classical Prikry forcing extension the family of the intermediate models can be parametrized by /finite. By modifying the standard Prikry tree forcing we define a Prikry-type forcing which also singularizes a measurable cardinal but which is minimal, i.e., there are no intermediate models properly between the ground model and the generic extension. The proof relies on combining the rigidity of the tree structure with indiscernibility arguments resulting from the normality of the associated measures.

In this paper, we investigate the existence of a Friedberg numbering in fragments of Peano Arithmetic and initial segments of Gödel's constructible hierarchy Lα, where α is Σ1 admissible. We prove that

(1) Over P− + BΣ2, the existence of a Friedberg numbering is equivalent to IΣ2, and

(2) For Lα, there is a Friedberg numbering if and only if the tame Σ2 projectum of α equals the Σ2 cofinality of α.

We show that any symmetric, Baire measurable function from the complement of E0 to a finite set is constant on an E0-nonsmooth square. A simultaneous generalization of Galvin's theorem that Baire measurable colorings admit perfect homogeneous sets and the Kanovei-Zapletal theorem canonizing Borel equivalence relations on E0-nonsmooth sets, this result is proved by relating E0-nonsmooth sets to embeddings of the complete binary tree into itself and appealing to a version of Hindman's theorem on the complete binary tree. We also establish several canonization theorems which follow from the main result.

In reverse mathematics, it is possible to have a curious situation where we know that an implication does not reverse, but appear to have no information on how to weaken the assumption while preserving the conclusion (other than reducing all the way to the tautology of assuming the conclusion). A main cause of this phenomenon is the proof of a sentence from the theory . Using methods based on the functional interpretation, we introduce a family of weakenings of and use them to give new upper bounds for the Nash-Williams Theorem of wqo theory and Menger's Theorem for countable graphs.

We prove that RCA0 + RRT ⊬ ACA0 where RRT is the Rainbow Ramsey Theorem for 2-bounded colorings of triples. This reverse mathematical result is based on a cone avoidance theorem, that every 2-bounded coloring of pairs admits a cone-avoiding infinite rainbow, regardless of the complexity of the given coloring. We also apply the proof of the cone avoidance theorem to the question whether RCA0 + RRT ⊦ ACA0 and obtain some partial answer.

We prove the consistency of together with the existence of a -definable mad family, answering a question posed by Friedman and Zdomskyy in [7, Question 16]. For the proof we construct a mad family in L which is an ℵ1-union of perfect a.d. sets, such that this union remains mad in the iterated Hechler extension. The construction also leads us to isolate a new cardinal invariant, the Borel almost-disjointness number, defined as the least number of Borel a.d. sets whose union is a mad family. Our proof yields the consistency of (and hence, ).

This paper investigates indifferent sets for comeager classes in Cantor space focusing of the class of all 1-generic sets and the class of all weakly 1-generic sets. Jockusch and Posner showed that there exist 1-generic sets that have indifferent sets [10]. Figueira, Miller and Nies have studied indifferent sets for randomness and other notions [7]. We show that any comeager class in Cantor space contains a comeager class with a universal indifferent set. A forcing construction is used to show that any 1-generic set, or weakly 1-generic set, has an indifferent set. Such an indifferent set can by computed by any set in which bounds the (weakly) 1-generic. We show by approximation arguments that some, but not all, 1-generic sets can compute an indifferent set for themselves. We show that all weakly 1-generic sets can compute an indifferent set for themselves. Additional results on indifferent sets, including one of Miller, and two of Fitzgerald, are presented.

We show a model construction for a system of higher-order illative combinatory logic thus establishing its strong consistency. We also use a variant of this construction to provide a complete embedding of first-order intuitionistic predicate logic with second-order propositional quantifiers into the system of Barendregt, Bunder and Dekkers, which gives a partial answer to a question posed by these authors.

Non-n-ampleness as denned by Pillay [20] and Evans [5] is preserved under analysability. Generalizing this to a more general notion of Σ-ampleness, this gives an immediate proof for all simple theories of a weakened version of the Canonical Base Property (CBP) proven by Chatzidakis [4] for types of finite SU-rank. This is then applied to the special case of groups.

We answer in the affirmative the following question of Jörg Brendle: If there is a Σ21 mad family, is there then a Π11 mad family?

In 1986, Osherson, Stob and Weinstein asked whether two variants of anomalous vacillatory learning, TxtFex** and TxtFext**, could be distinguished [3]. In both, a machine is permitted to vacillate between a finite number of hypotheses and to make a finite number of errors. TxtFext**-learning requires that hypotheses output infinitely often must describe the same finite variant of the correct set, while TxtFex**-learning permits the learner to vacillate between finitely many different finite variants of the correct set. In this paper we show that TxtFex** ≠ TxtFext**, thereby answering the question posed by Osherson, et al. We prove this in a strong way by exhibiting a family in TxtFex*2 TxtFext**.

A generic computation of a subset A of ℕ consists of a computation that correctly computes most of the bits of A, and never incorrectly computes any bits of A, but which does not necessarily give an answer for every input. The motivation for this concept comes from group theory and complexity theory, but the purely recursion theoretic analysis proves to be interesting, and often counterintuitive. The primary result of this paper is that there are no minimal pairs for generic computability, answering a question of Jockusch and Schupp.

BD-N is a weak principle of constructive analysis. Several interesting principles implied by BD-N have already been identified, namely the closure of the anti-Specker spaces under product, the Riemann Permutation Theorem, and the Cauchyness of all partially Cauchy sequences. Here these are shown to be strictly weaker than BD-N, yet not provable in set theory alone under constructive logic.

A pointwise definable model is one in which every object is definable without parameters. In a model of set theory, this property strengthens V = HOD, but is not first-order expressible. Nevertheless, if ZFC is consistent, then there are continuum many pointwise definable models of ZFC. If there is a transitive model of ZFC, then there are continuum many pointwise definable transitive models of ZFC. What is more, every countable model of ZFC has a class forcing extension that is pointwise definable. Indeed, for the main contribution of this article, every countable model of Gödel-Bernays set theory has a pointwise definable extension, in which every set and class is first-order definable without parameters.

We answer a question of Cummings and Magidor by proving that the P-ideal dichotomy of Todorčević refutes □κ,ω for any uncountable κ. We also show that the P-ideal dichotomy implies the failure of □κ,<b provided that cf(κ) > ω1.

We give an example of an imaginary defined in certain valued fields with analytic structure which cannot be coded in the ‘geometric’ sorts which suffice to code all imaginaries in the corresponding algebraic setting.

The half-open real unit interval (0,1] is closed under the ordinary multiplication and its residuum. The corresponding infinite-valued propositional logic has as its equivalent algebraic semantics the equational class of cancellative hoops. Fixing a strong unit in a cancellative hoop—equivalently, in the enveloping lattice-ordered abelian group—amounts to fixing a gauge scale for falsity. In this paper we show that any strong unit in a finitely presented cancellative hoop H induces naturally (i.e., in a representation-independent way) an automorphism-invariant positive normalized linear functional on H. Since H is representable as a uniformly dense set of continuous functions on its maximal spectrum, such functionals—in this context usually called states—amount to automorphism-invariant finite Borel measures on the spectrum. Different choices for the unit may be algebraically unrelated (e.g., they may lie in different orbits under the automorphism group of H), but our second main result shows that the corresponding measures are always absolutely continuous w.r.t. each other, and provides an explicit expression for the reciprocal density.

For a countable structure , the spectrum is the set of Turing degrees of isomorphic copies of . For a complete elementary first order theory T, the spectrum is the set of Turing degrees of models of T. We answer a question from [1] by showing that there is an atomic theory T whose spectrum does not match the spectrum of any structure.