We establish a coloring theorem for successors of singular cardinals, and use it prove that for any such cardinal μ, we have if and only if for arbitrarily large θ < μ.

We study definability of second order generalized quantifiers on finite structures. Our main result says that for every second order type t there exists a second order generalized quantifier of type t which is not definable in the extension of second order logic by all second order generalized quantifiers of types lower than t.

In this paper we answer the question of whether all low sets with the splitting property are promptly simple. Further we try to make the role of lowness properties and prompt simplicity in the construction of automorphisms of the lattice of r.e. (recursively enumerable) sets more perspicuous. It turns out that two new properties of r.e. sets, which are dual to each other, are essential in this context: the prompt and the low shrinking property.

In an earlier paper [4] we had shown (using Soare's automorphism construction [10] and [12]) that all r.e. generic sets are automorphic in the lattice ℰ of r.e. sets under inclusion. We called a set A promptly simple if Ā is infinite and there is a recursive enumeration of A and the r.e. sets (We)e∈N such that if We is infinite then there is some element (or equivalently: infinitely many elements) x of We such that x gets into A “promptly” after its appearance in We (i.e. for some fixed total recursive function f we have x ∈ Af(s), where s is the stage at which x entered We). Prompt simplicity in combination with lowness turned out to capture those properties of r.e. generic sets that were used in the mentioned automorphism result. In a following paper with Shore and Stob [7] we studied an ℰ-definable consequence of prompt simplicity: the splitting property.

Using a result of Gurevich and Lewis on the word problem for finite semigroups, we give short proofs that the following theories are hereditarily undecidable: (1) finite graphs of vertex-degree at most 3; (2) finite nonvoid sets with two distinguished permutations; (3) finite-dimensional vector spaces over a finite field with two distinguished endomorphisms.

We analyse the Mitchell ordering in a model where κ is -hypermeasurable and > .

In [3], Todorčević showed that ω1 ⇸ [ω1]ω12. In this paper we use similar methods to prove an analogous partition theorem for Pω1(λ), for certain uncountable cardinals λ.

Recall that ω1 → [ω1]ω12, means that for every function f: [ω1]2 → ω1 there is a set A ∈ [ω1]ω1 such that f“[A]2 ≠ ω1, and of course ω1 ⇸ [ω1]ω12, is the negation of this statement. For partition relations on Pω1(→) it is customary to partition only those pairs of sets in which the first set is a subset of the second. Thus for A ⊆ Pω1(λ) we define

We will write Pω1(λ) → [unbdd]λ2 to mean that for every function f: [Pω1(λ)]⊂2 → λ there is an unbounded set A ⊆ Pω1(λ) such that f“[A]⊂2 ≠ λ, and again Pω1(λ) ⇸ [unbdd]λ2 is the negation of this statement.

We show that all QE rings of prime power characteristic are constructed in a straightforward way out of three components: a filtered Boolean power of a finite field, a nilpotent Jacobson radical, and the ring Zp. or the Witt ring W2(F4) (which is the characteristic four analogue of the Galois field with four elements).

Theorem. Assume that there exists a standard model of ZFC + V = L. Then there is a model of ZFC in which the partial ordering of the degrees of constructibility of reals is isomorphic with a given finite lattice.

The proof of the theorem uses forcing. The definition of the forcing conditions and the proofs of some of the lemmas are connected with Lerman's paper on initial segments of the upper semilattice of the Turing degrees [2]. As an auxiliary notion we shall introduce the notion of a sequential representation of a lattice, which slightly differs from Lerman's representation.

Let K be a given finite lattice. Assume that the universe of K is an integer l. Let ≤ K be the ordering in K. A sequential representation of K is a sequence Ui ⊆ Ui+1 of finite subsets of ω i such that the following holds:

(1) For any s, s′ Є Ui , i Є ω, k, m Є l, k ≤ K m & s(m) = s′(m) → s(k) = s′(k).

(2) For any s Є Ui , i Є ω, s(0) = 0 where 0 is the least element of K.

(3) For any s, s′ Є i Є ω, k,j Є l, if k y Kj = m and s(k) = s′(k) & s(j) = s′(j) → s(m) = s′(m), where v K denotes the join in K.

An Ehrenfeucht theory is a complete first order theory with exactly n countable models up to isomorphism, 1 < n < ω. Numerous results have emerged regarding these theories ([1]–[15]). A general question in model theory is whether or not the number of countable models of a complete theory can be different than the number of countable models of a complete consistent extension of the theory by finitely many constant symbols. Examples are known of Ehrenfeucht theories that have complete extensions by finitely many constant symbols such that the extensions fail to be Ehrenfeucht ([4], [8], [13]). These examples are easily modified to allow finite increases in the number of countable models.

This paper contains examples in the other direction—complete theories that have consistent extensions by finitely many constant symbols such that the extensions have fewer countable models. This answers affirmatively a question raised by, among others, Peretyat'kin [8]. The first example will be an Ehrenfeucht theory with exactly four countable models with an extension by a constant symbol that has only three countable models. The second example will be a complete theory that is not Ehrenfeucht, but which has an extension by a constant symbol that is Ehrenfeucht. The notational conventions for this paper are standard.

Peretyat'kin introduced the theory of a dense binary branching tree with a meet operator [7]. Dense ω-branching trees have also proven useful [5], [11]. Both of the Theories that will be constructed make use of dense ω-branching trees.

Hilbert and Ackermann ([1], p. 107) define a first order axiom system as one in which the axioms contain one or more predicate constants, but no predicate variables. Here “axiom” refers to the specific subject-matter axioms and not to the rules of the restricted predicate calculus (quantification theory), which rules are presupposed for each first-order system. It is pointed out by them that an exception could be made for the predicate of identity; for the axiom scheme for this predicate, namely

which has in (b) the variable predicate F, could nevertheless be replaced, in any given first-order system, by a finite set of axioms without predicate variables. Thus, for example, if Φ[x, y) is the one constant predicate of such a system then PId(b) could be replaced by

Thus one postulates, in addition to the reflexivity, symmetry, and transitivity of identity, the substitutivity of identical entities in each of the possible “atomic” contexts of a variable (occurrences in the primitive predicates). In this method of introducing identity it has to be taken as an additional primitive predicate and further axioms are consequently needed. In such a system having PId(a), (b1)−(b3) as axioms, the scheme PId(b) can be derived as a meta-theorem of the system, F(x) then being any formula of the system.

The principal aim of this paper is to present a construction method for nontotal extensional combinatory algebras. This is done in §2. In §0 we give definitions of some basic notions for partial combinatory algebras from which the corresponding notions for (total) combinatory algebras are obtained as specializations. In §1 we discuss some properties of nontotal extensional combinatory algebras in general. §2 describes a “partial” variant of reflexive complete partial orders yielding nontotal extensional combinatory algebras. Finally, §3 deals with properties of the models constructed in §2, such as incompletability, having no total submodel and the pathological behaviour with respect to the interpretation of unsolvable λ-terms.

We show that the polynomial translation of the classical propositional normal modal logic S4 into the intuitionistic propositional logic Int from Fernández is incorrect. We give a modified translation and prove its correctness, and provide implementations of both translations to allow others to test our results.

This paper is a continuation of [27], where we provide the background and the basic tools for studying the structural properties of classes of models over languages without equality. In the context of such languages, it is natural to make distinction between two kinds of classes, the so-called abstruct classes, which correspond to those closed under isomorphic copies in the presence of equality, and the reduced classes, i.e., those obtained by factoring structures by their largest congruences. The generic problem described in [27] is to investigate under what conditions this reduction process does not alter the metatheory of a class.

Here we focus our attention on a concrete aspect of this generic problem that we import from universal algebra, namely the existence and description of free models. As in [27], we can find here again the basic notion of protoalgebraicity, which was originally introduced in [7] as the weakest condition to guarantee that the reduction process behaves reasonably well from an algebraic point of view. Our concern, however, takes us to handle a further notion, that of semialgebraicity, which corresponds to the notion of equivalential logic of [18]; semialgebraicity turns out to be the property which ensures that freeness is fully preserved by the reduction process.

In a 1969 paper, Quine coined the term ‘limits of decision”. This term evidently refers to limits on the logical vocabulary of a logic, beyond which satisfiability is no longer decidable. In the same paper, Quine showed that not only monadic formulas, but homogeneous k -adic formulas for arbitrary k lie on the decidable side of the limits of decision. But the precise location of the limits of decision has remained an open question. The present paper answers that question. It addresses the question of decidability of those sublogics of first-order logic that are defined in terms of their logical vocabularies. A complete answer is obtained, thus locating exactly Quine's limits of decision.

Criteria are obtained for a filter ${\cal F}$ of subsets of a set I to be an intersection of finitely many ultrafilters, respectively, finitely many κ-complete ultrafilters for a given uncountable cardinal $\kappa .$ From these, general results are deduced concerning homomorphisms on infinite direct product groups, which yield quick proofs of some results in the literature: the Łoś–Eda theorem (characterizing homomorphisms from a not-necessarily-countable direct product of modules to a slender module), and some results of Nahlus and the author on homomorphisms on infinite direct products of not-necessarily-associative k-algebras. The same tools allow other results of Nahlus and the author to be nontrivially strengthened, and yield an analog to one of their results, with nonabelian groups taking the place of k-algebras.

We briefly examine the question of how the common technique used in applying the general results of this note to k-algebras on the one hand, and to nonabelian groups on the other, might be extended to more general varieties of algebras in the sense of universal algebra.

In a final section, the Erdős–Kaplansky theorem on dimensions of vector spaces ${D^I}$$(D$ a division ring) is extended to reduced products ${D^I}/{\cal F},$ and an application is noted.

In this paper, we will present a definability theorem for first order logic. This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if M is a model of a theory T in a language , then, clearly, any definable subset S ⊂ M (i.e., a subset S = {a ∣ M ⊨ φ(a)} defined by some formula φ) is invariant under all automorphisms of M. The same is of course true for subsets of Mn defined by formulas with n free variables.

Our theorem states that, if one allows Boolean valued models, the converse holds. More precisely, for any theory T we will construct a Boolean valued model M, in which precisely the T -provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is definable by a formula .

Our presentation is entirely selfcontained, and only requires familiarity with the most elementary properties of model theory. In particular, we have added a first section in which we review the basic definitions concerning Boolean valued models.

If A and B are subsets of natural numbers we say that A is recursively equivalent to B (denoted A ≃ B) if there is a one-one partial recursive function which maps A onto B, and that A is recursively isomorphic to B (denoted A ≅ B) if there is a one-one total recursive function which maps A onto B and Ā (the complement of A) onto B#x00AF;.

We present abstract complexity results about Coquand and Hyland–Ong game semantics, that will lead to new bounds on the length of first-order cut-elimination, normalization, interaction between expansion trees and any other dialogical process game semantics can model and apply to. In particular, we provide a novel method to bound the length of interactions between visible strategies and to measure precisely the tower of exponentials defining the worst-case complexity. Our study improves the old estimates on average by several exponentials.