We give an example of a countable theory T such that for every cardinal λ ≥ ℵ2 there is a fully indiscernible set A of power λ such that the principal types are dense over A, yet there is no atomic model of T over A. In particular, T(A) is a theory of size λ where the principal types are dense, yet T(A) has no atomic model.

We say that a ring admits elimination of quantifiers, if in the language of rings, {0, 1, +, ·}, the complete theory of R admits elimination of quantifiers.

Theorem 1. Let D be a division ring. Then D admits elimination of quantifiers if and only if D is an algebraically closed or finite field.

A ring is prime if it satisfies the sentence: ∀x∀y∃z (x =0 ∨ y = 0∨ xzy ≠ 0).

Theorem 2. If R is a prime ring with an infinite center and R admits elimination of quantifiers, then R is an algebraically closed field.

Let be the class of finite fields. Let be the class of 2 × 2 matrix rings over a field with a prime number of elements. Let be the class of rings of the form GF(pn)⊕GF(pk) such that either n = k or g.c.d. (n, k) = 1. Let be the set of ordered pairs (f, Q) where Q is a finite set of primes and such that the characteristic of the ring f(q) is q. Finally, let be the class of rings of the form ⊕q ∈ Qf(q), for some (f, Q) in .

Theorem 3. Let R be a finite ring without nonzero trivial ideals. Then R admits elimination of quantifiers if and only if R belongs to.

Theorem 4. Let R be a ring with the descending chain condition of left ideals and without nonzero trivial ideals. Then R admits elimination of quantifiers if and only if R is an algebraically closed field or R belongs to.

In contrast to Theorems 2 and 4, we have

Theorem 5. If R is an atomless p-ring, then R is finite, commutative, has no nonzero trivial ideals and admits elimination of quantifiers, but is not prime and does not have the descending chain condition.

We also generalize Theorems 1, 2 and 4 to alternative rings.

Resolution is the most widely studied approach to propositional theorem proving. In developing efficient resolution-based algorithms, dozens of variants and refinements of resolution have been studied from both the empirical and analytic sides. The most prominent of these refinements are: DP (ordered), DLL (tree), semantic, negative, linear and regular resolution. In this paper, we characterize and study these six refinements of resolution. We give a nearly complete characterization of the relative complexities of all six refinements. While many of the important separations and simulations were already known, many new ones are presented in this paper; in particular, we give the first separation of semantic resolution from general resolution. As a special case, we obtain the first exponential separation of negative resolution from general resolution. We also attempt to present a unifying framework for studying all of these refinements.

We prove that an intermediate predicate logic characterized by a class of finite partially ordered sets is recursively axiomatizable iff it is “finite”, i.e., iff it is characterized by a single finite partially ordered set. Therefore, the predicate logic LFin of the class of all predicate Kripke frames with finitely many possible worlds is not recursively axiomatizable.

This paper is concerned with recursive structures and the persistance of an effective notion of categoricity. The terminology and notational conventions are standard. We will devote the rest of this paragraph to a cursory review of some of the assumed conventions. If θ is a formula, then θk denotes θ if k is zero, and ¬θ if k is one. If A is a sequence with domain a subset of ω, then A∣ n denotes the sequence obtained by restricting the domain of A to n. For an effective first order language L, let {ci ∣i<ω} be distinct new constants, and let {θ i ∣i<ω} be an effective enumeration of all sentences in the language L ∪ {ci ∣j<ω}. An infinite L-structure is recursive iff it has universe ω and the set is recursive, where cn is interpreted by n. In general we say that a set of formulas is recursive if the set of its indices with respect to an enumeration such as above is recursive. The ∃-diagram of a structure is recursive if the structure is recursive and the set and θ i is an existential sentence} is also recursive. The definition of “the ∀∃-diagram of is recursive” is similar.

We study when classes can have the disjoint amalgamation property for a proper initial segment of cardinals.

For every natural number k, there is a class Kk, defined by a sentence in Lω1,ω that has no models of cardinality greater than ℶk + 1, but Kk has the disjoint amalgamation property on models of cardinality less than or equal to ℵk − 3 and has models of cardinality ℵk − 1.

More strongly, we can have disjoint amalgamation up to ℵ∝ for ∝ < ω1, but have a bound on size of models.

For every countable ordinal ∝, there is a class K∝ defined by a sentence in Lω1,ω that has no models of cardinality greater than ℶω1, but K does have the disjoint amalgamation property on models of cardinality less than or equal to ℵ∝.

Finally we show that we can extend the ℵ∝ to ℶ∝ in the second theorem consistently with ZFC and while having ℵi ≪ ℶi for 0 < i < ∝. Similar results hold for arbitrary ordinals ∝ with ∣∝∣ = k and Lk + ω.

We show that for any real number, the class of real numbers less random than it, in the sense of rK-reducibility, forms a countable real closed subfield of the real ordered field. This generalizes the well-known fact that the computable reals form a real closed field.

With the same technique we show that the class of differences of computably enumerable reals (d.c.e. reals) and the class of computably approximable reals (c.a. reals) form real closed fields. The d.c.e. result was also proved nearly simultaneously and independently by Ng (Keng Meng Ng, Master's Thesis, National University of Singapore, in preparation).

Lastly, we show that the class of d.c.e. reals is properly contained in the class or reals less random than Ω (the halting probability), which in turn is properly contained in the class of c.a. reals, and that neither the first nor last class is a randomness class (as captured by rK-reducibility).

We prove that the set of formulae provable in the full Lambek calculus with the structural rule of contraction is undecidable. In fact, we show that the positive fragment of this logic is undecidable.

In this note we study generic existence of maximal almost disjoint (MAD) families. Among other results we prove that Cohen-indestructible families exist generically if and only if b = c. We obtain analogous results for other combinatorial properties of MAD families, including Sacks-indestructibility and being +-Ramsey.

We prove that a wide class of strongly proper forcing posets have quotients with strong properties. Specifically, we prove that quotients of forcing posets which have universal strongly generic conditions on a stationary set of models by certain nice regular suborders satisfy the ω1-approximation property. We prove that the existence of stationarily many ω1-guessing models in Pω2(H(θ)), for sufficiently large cardinals θ, is consistent with the continuum being arbitrarily large, solving a problem of Viale and Weiss [13].

If a and b are degrees of unsolvability, a is called (as in [1]) a minimal cover of b if b < a and no degree c satisfies b < c < a. A degree a is called a minimal cover if it is a minimal cover of some degree b. We show in ZF set theory that there is a degree d such that every degree a ≥ d is a minimal cover. In [1] it was remarked that this result follows via a lemma of D. A. Martin [4] from the determinacy of a certain Σ50 Gale-Stewart game (which in turn follows [5] from the existence of a measurable cardinal). The argument here is parallel to that of [1], but the essential new ingredient is the result of Paris [6] that Σ50 determinacy can be proved in ZF. Also it is necessary to replace the Σ50 game of [1] by a related Σ40 game and to use the method, rather than just the statement, of Martin's lemma.

Theorem. There is a degreedsuch that every degreea ≥ dis a minimal cover.

Proof. In a (Gale–Stewart) game on 2ω, players I and II construct a set of integers A as follows: At the nth stage, player I determines whether n ϵ A if n is even and player II if n is odd. Thus player I controls A0 and player II controls A1 where Ai = {n: 2n + i ϵ A}. Let Aij denote (Ai)j and let M (A, B) mean that the degree of A is a minimal cover of that of B. Consider now the particular game in which player I wins iff M(A, A00) holds.

In this paper we develop certain methods of proof in Quine's set theory NF which have no counterparts elsewhere. These ideas were first used by Specker [5] in his disproof of the Axiom of Choice in NF. They depend on the properties of two related operations, T(n) on cardinal numbers and U(α) on ordinal numbers, which are defined by the equations

for each set x and well ordering R. (Here and below we use Rosser's notation [3].) The definitions insure that the formulas T(x) = y and U(x) = y are stratified when y is assigned a type one higher than x. The importance of T and U stems from the following facts: (i) each of T and U is a 1-1, order preserving operation from its domain onto a proper initial section of its domain; (ii) Tand U commute with most of the standard operations on cardinal and ordinal numbers.

These basic facts are discussed in §1. In §2 we prove in NF that the exponential function 2n is not 1-1. Indeed, there exist cardinal numbers m and n which satisfy

In §3 we prove the following technical result, which has many important applications. Suppose f is an increasing function from an initial segment S of the set NO of ordinal numbers into NO and that f commutes with U.

In this work we investigate the Weihrauch degree of the problem Decreasing Sequence ( $\mathsf {DS}$ ) of finding an infinite descending sequence through a given ill-founded linear order, which is shared by the problem Bad Sequence ( $\mathsf {BS}$ ) of finding a bad sequence through a given non-well quasi-order. We show that $\mathsf {DS}$ , despite being hard to solve (it has computable inputs with no hyperarithmetic solution), is rather weak in terms of uniform computational strength. To make the latter precise, we introduce the notion of the deterministic part of a Weihrauch degree. We then generalize $\mathsf {DS}$ and $\mathsf {BS}$ by considering $\boldsymbol {\Gamma }$ -presented orders, where $\boldsymbol {\Gamma }$ is a Borel pointclass or $\boldsymbol {\Delta }^1_1$ , $\boldsymbol {\Sigma }^1_1$ , $\boldsymbol {\Pi }^1_1$ . We study the obtained $\mathsf {DS}$ -hierarchy and $\mathsf {BS}$ -hierarchy of problems in comparison with the (effective) Baire hierarchy and show that they do not collapse at any finite level.

Let A2 be the axiomatic system of second order arithmetic as described in [2]. One of the models of A2 is the “principal model” Mpr consisting of all integers and all sets of integers. Obviously there exist many denumerable ω-models elementarily equivalent to Mpr and we shall deal in this paper with some questions pertaining to this family which we denote by .

In §1 we define a rather natural relation ε between two denumerable families of sets of integers. From the upward Skolem-Löwenheim theorem it follows easily that there exists a family ordered by ε in the type ω1, but it is not immediately obvious whether there exist a subfamily of not well-ordered by ε. In the present paper we construct such a family of type η. ω1 where η is the order type of rationals and indicate some applications to hyperdegrees.

We adopt the terminology and notation of [2], with the only change that we adjoin to the language of A2 the constants ν0, ν1, … for the consecutive numerals 0, 1, 2, … and axioms which characterise them:

Also we modify the axioms of A2 given in [2] by prefixing them by general quantifiers bounded either to S or to N.

We show, for each computable ordinal α and degree a > 0(α), the existence of a torsion-free abelian group with proper αth jump degree a.

We investigate some relations between omitting types of a countable theory and some notions defined in terms of the real line, such as for example the ideal of meager subsets of R. We also try to express connections between the logical structure of a theory and the existence of its countable models omitting certain families of types.

It is well known that assuming MA we can omit < nonisolated types. But MA is rather a strong axiom. We prove that in order to be able to omit < nonisolated types it is sufficient to assume that the real line cannot be covered by less than meager sets; and this is in fact the weakest possible condition. It is worth pointing out that by means of forcing we can easily obtain the model of ZFC in which R cannot be covered by < meager sets. It suffices to add to the ground model Cohen generic reals.

We also formulate similar results for omitting pairwise contradictory types. It turns out that from some point of view it is much more difficult to find the family of pairwise contradictory types which cannot be omitted by a model of T, than to find such a family of possibly noncontradictory types. Moreover, for any two countable theories T1, T2 without prime models, the existence of a family of κ types which cannot be omitted by a model of T1 is equivalent to the existence of such a family for T2. This means that from the point of view of omitting types all theories without prime models are identical. Similar results hold for omitting pairwise contradictory types.

In [1, §4], my ‘proof’ that GH1 has the finite model property is incorrect; there are considerable obscurities towards the end of §1, particularly on p. 33; and I should have exhibited the finite models for GH1. In §1 of this paper I expand the analysis of the sub-directly irreducible models for GH1 which I give in §1 of [1]. In §2 I give a correct proof that GH1 has the finite model property. In §3 I exhibit these finite models as models on certain ordered sets.

In this paper we prove three theorems about first-order theories that are categorical in a higher power. The first theorem asserts that such a theory either is totally categorical or there exist prime and minimal models over arbitrary base sets. The second theorem shows that such theories have a natural notion of dimension that determines the models of the theory up to isomorphism. From this we conclude that I(T,ℵα,) = ℵ0 + ∣α∣ where ℵα = the number of formulas modulo T-equivalence provided that T is not totally categorical. The third theorem gives a new characterization of these theories.