Beginning with Ramsey's theorem of 1930, combinatorists have been intrigued with the notion of large cardinals satisfying partition relations. Years of research have established the smaller ones, weakly ineffable, Erdös, Jonsson, Rowbottom and Ramsey cardinals to be among the most interesting and important large cardinals in set theory. Recently, cardinals satisfying more powerful infinite-exponent partition relations have been examined with growing interest. This is due not only to their inherent qualities and the fact that they imply the existence of other large cardinals (Kleinberg [2], [3]), but also to the fact that the Axiom of Determinacy (AD) implies the existence of great numbers of such cardinals (Martin [5]).

That these properties are more often than not inconsistent with the full Axiom of Choice (Kleinberg [4]) somewhat increases their charm, for the theorems concerning them tend to be a little odd, and their proofs, circumforaneous. The properties are, as far as anyone knows, however, consistent with Dependent Choice (DC).

Our basic theorem will be the following: If k > ω and k satisfies k→(k)k then the least cardinal δ such that has a δ-additive, uniform ultrafilter. In addition, if ACω is assumed, we will show that δ is greater than ω, and hence a measurable cardinal. This result will be strengthened somewhat when we prove that for any k, δ, if then .

Let T be a complete first-order theory over a finite relational language which is axiomatized by universal and existential sentences. It is shown that T is almost trivial in the sense that the universe of any model of T can be written . where F is finite and I 1, I 2, …, In are mutually indiscernible over F. Some results about complete theories with ∃∀-axioms over a finite relational language are also mentioned.

We study the connection between Schnorr triviality and genericity. We show that while no 2-generic is Turing equivalent to a Schnorr trivial and no 1-generic is tt-equivalent to a Schnorr trivial, there is a 1-generic that is Turing equivalent to a Schnorr trivial. However, every such 1-generic must be high. As a corollary, we prove that not all K-trivials are Schnorr trivial. We also use these techniques to extend a previous result and show that the bases of cones of Schnorr trivial Turing degrees are precisely those whose jumps are at least 0″.

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.

A fundamental problem of symbolic logic is to define logical calculi sufficient to comprise important parts of mathematics, and to develop systematic methods of calculation therein.

The possibility of progress in this direction has been severely limited by Gödel's proof that a consistent system sufficient to comprise arithmetic must contain propositions whose truth-value cannot be decided within the system, and by Church's extension of Gödel's method to the result that even in the first order logical function calculus the general decision problem cannot be solved.

A significant portion of the study of large cardinals in set theory centers around the concept of “partition relation”. To best capture the basic idea here, we introduce the following notation: for x and y sets, κ an infinite cardinal, and γ an ordinal less than κ, we let [x]γ denote the collection of subsets of x of order-type γ and abbreviate with the partition relation for each function F frominto y there exists a subset C of κ of cardinality κ such that (such that for each α < γ) the range of F on [С]γ ([С]α) has cardinality 1. Now although each infinite cardinal κ satisfies the relation for each n and m in ω (F. P. Ramsey [8]), a connection with large cardinals arises when one asks, “For which uncountable κ do we have κ → (κ)2?” Indeed, any uncountable cardinal κ which satisfies κ → (κ)2 is strongly inaccessible and weakly compact (see [9]). As another example one can look at the improvements of Scott's original result to the effect that if there exists a measurable cardinal then there exists a nonconstructible set. Indeed, if κ is a measurable cardinal then κ → (κ)< ω, and as Solovay [11] has shown, if there exists a cardinal κ such that κ → (κ)< ω3 (κ → (ℵ1)< ω, even) then there exists a nonconstructible set of integers.

In 2004 Csima, Hirschfeldt, Knight, and Soare [1] showed that a set A ≤T 0′ is nonlow2 if and only if A is prime bounding, i.e., for every complete atomic decidable theory T, there is a prime model computable in A. The authors presented nine seemingly unrelated predicates of a set A, and showed that they are equivalent for sets. Some of these predicates, such as prime bounding, and others involving equivalence structures and abelian p-groups come from model theory, while others involving meeting dense sets in trees and escaping a given function come from pure computability theory.

As predicates of A, the original nine properties are equivalent for sets; however, they are not equivalent in general. This article examines the (degree-theoretic) relationship between the nine properties. We show that the nine properties fall into three classes, each of which consists of several equivalent properties. We also investigate the relationship between the three classes, by determining whether or not any of the predicates in one class implies a predicate in another class.

A. N. Prior in [9] has suggested an approach towards a second order logic of existence where, following medieval logicians, we distinguish “between predicates (like ‘is red’, ‘is hard’, etc.) which entail existence, and predicates (like ‘is thought to be red’, ‘is thought of’, etc.) which do not.”2 Let us refer to attributes (including relational attributes) which are designated by the former kind of predicate as existence attributes, or for brevity, e-attributes.

The degree spectrum of a countable structure is the set of all Turing degrees of presentations of that structure. We show that every nonlow Turing degree lies in the spectrum of some differentially closed field (of characteristic 0, with a single derivation) whose spectrum does not contain the computable degree 0. Indeed, this is an equivalence, for we also show that if this spectrum contained a low degree, then it would contain the degree 0. From these results we conclude that the spectra of differentially closed fields of characteristic 0 are exactly the jump-preimages of spectra of automorphically nontrivial graphs.

This paper investigates the principles that one must add to Boolean algebra to capture reasoning not only about intersection, union, and complementation of sets, but also about the relative size of sets. We completely axiomatize such reasoning under the Cantorian definition of relative size in terms of injections.

It will be shown later in this paper that the class of all Hilbert spaces is not an elementary class (in the wider sense) in the lower predicate calculus. It is not difficult to find a type and a sentence in Lω1ω1 (of that type) whose models are precisely the Hilbert spaces (slightly altered to include in their domains the real or complex numbers).

Finitely axiomatizable ℵ1 categorical theories are locally modular.

We analyze combinatorial properties of open covers of sets of real numbers by using filters on the natural numbers. In fact, the goal of this paper is to characterize known properties related to ω-covers of the space in terms of combinatorial properties of filters associated with these ω-covers. As an example, we show that all finite powers of a set of real numbers have the covering property of Menger if, and only if, each filter on ω associated with its countable ω-cover is a P+ filter.

In this paper, we use algebra-valued models to study cardinal numbers in a class of non-classical set theories. The algebra-valued models of these non-classical set theories validate the Axiom of Choice, if the ground model validates it. Though the models are non-classical, the foundations of cardinal numbers in these models are similar to those in classical set theory. For example, we show that mathematical induction, Cantor’s theorem, and the Schröder–Bernstein theorem hold in these models. We also study a few basic properties of cardinal arithmetic. In addition, the generalized continuum hypothesis is proved to be independent of these non-classical set theories.

We use finite model theory (in particular, the method of FM-truth definitions, introduced in [MM01] and developed in [K04], and a normal form result akin to those of [Ste93] and [G97]) to prove:

Let m ≥ 2. Then:

(A) If there exists k such that NP⊆ Σm TIME(nk)∩ Πm TIME(nk), then for every r there exists kr such that :

(B) If there exists a superpolynomial time-constructible function f such that NTIME(f), then additionally .

This strengthens a result by Mocas [M96] that for any r, .

In addition, we use FM-truth definitions to give a simple sufficient condition for the arity hierarchy to be strict over finite models.

The results that follow are intended to be understood as informal counterparts to formal theorems of Zermelo-Fraenkel set theory with choice. Basic notation not explained here can usually be found in [5]. It will also be necessary to assume a knowledge of the fundamentals of boolean and generic extensions, in the style of Jech's monograph [3]. Consistency results will be stated as assertions about the existence of certain complete boolean algebras, B, C , etc., either outright or in the sense of a countable standard transitive model M of ZFC augmented by hypotheses about the existence of various large cardinals. Proofs will usually be phrased in terms of the forcing relation ⊩ over such an M, especially when they make heavy use of genericity. They are then assertions about Shoenfield-style P -generic extensions M(G), in which the ‘names’ are required without loss of generality to be elements of M B = (V B )M, B is the boolean completion of P in M (cf. [3, p. 50]: the notation there is RO(P)), the generic G is named by Ĝ ∈ M B such that (⟦p ∈ Ĝ⟧ B = p and (cf. [11, p. 361] and [3, pp. 58–59]), and for p ∈ P and c 1, …, cn ∈ M B , p ⊩ φ(c 1, …, cn ) iff ⟦φ(c 1, …, cn )⟧B ≥ p (cf. [3, pp. 61–62]).

Some prior acquaintance with large cardinal theory is also needed. At this writing no comprehensive introductory survey is yet in print, though [1], [10], [12]and [13] provide partial coverage. The scheme of definitions which follows is intended to fix notation and serve as a glossary for reference, and it is followed in turn by a description of the results of the paper. We adopt the convention that κ, λ, μ, ν, ρ and σ vary over infinite cardinals, and all other lower case Greek letters (except χ, φ, ψ, ϵ) over arbitrary ordinals.

We investigate the relationship between lowness for Schnorr randomness and Schnorr triviality. We show that a real is low for Schnorr randomness if and only if it is Schnorr trivial and hyperimmune free.

We show that Diophantine equivalence of two suitably presented countable rings implies that the existential polynomial languages of the two rings have the same “expressive power” and that their Diophantine sets are in some sense the same. We also show that a Diophantine class of countable rings is contained completely within a relative enumeration class and demonstrate that one consequence of this fact is the existence of infinitely many Diophantine classes containing holomorphy rings of ℚ.