In this paper we present a new proof of a decidability result for the firstorder theories of certain subvarieties of Heyting algebras. By a famous result of Grzegorczyk, the full first-order theory of Heyting algebras is undecidable. In contrast, the first-order theory of Boolean algebras and of many interesting subvarieties of Boolean algebras is decidable by a result of Tarski [8]. In fact, Kozen [6] gives a comprehensive quantitative classification of the complexities of the first-order theories of various subclasses of Boolean algebras (including the full variety).

This stark contrast may be reconciled from the standpoint of universal algebra as arising out of the byplay between structure and decidability: A good structure theory entails positive decidability results. Boolean algebras have a well-developed structure theory [5], while the corresponding theory for Heyting algebras is quite meagre. Viewed in this way, we may hope to obtain decidability results if we focus attention on subclasses of Heyting algebras with good structural properties.

K. Idziak and P. M. Idziak [4] have considered an interesting subvariety of Heyting algebras, , which is the variety generated by all linearly-ordered Heyting algebras. This variety is shown to be the largest subvariety of Heyting algebras with a decidable theory of its finite members. However their proof is rather indirect, proceeding via semantic interpretation into the monadic second order theory of trees. The latter is a powerful theory—it interprets many other theories—but is computationally highly infeasible. In fact, by a celebrated theorem of Rabin, its complexity is not bounded by any elementary recursive function. Consequently, the proof of [4], besides being indirect, also gives no information on the quantitative computational complexity of the theory of .

Here we pursue the theme of structure and decidability. We isolate the indecomposable algebras in and use this to prove a theorem on the structure of if -algebras. This theorem relates the -algebras structurally to Boolean algebras. This enables us to bootstrap the known decidability results for Boolean algebras to the variety if .

The Schur-Zassenhaus Theorem is one of the fundamental theorems of finite group theory. Here is its statement:

Fact 1.1 (Schur-Zassenhaus Theorem). Let G be a finite group and let N be a normal subgroup of G. Assume that the order ∣N∣ is relatively prime to the index [G:N]. Then N has a complement in G and any two complements of N are conjugate in G.

The proof can be found in most standard books in group theory, e.g., in [S, Chapter 2, Theorem 8.10]. The original statement stipulated one of N or G/N to be solvable. Since then, the Feit-Thompson theorem [FT] has been proved and it forces either N or G/N to be solvable. (The analogous Feit-Thompson theorem for groups of finite Morley rank is a long standing open problem).

The literal translation of the Schur-Zassenhaus theorem to the finite Morley rank context would state that in a group G of finite Morley rank a normal π-Hall subgroup (if it exists at all) has a complement and all the complements are conjugate to each other. (Recall that a group H is called a π-group, where π is a set of prime numbers, if elements of H have finite orders whose prime divisors are from π. Maximal π-subgroups of a group G are called π-Hall subgroups. They exist by Zorn's lemma. Since a normal π-subgroup of G is in all the π-Hall subgroups, if a group has a normal π-Hall subgroup then this subgroup is unique.)

The second assertion of the Schur-Zassenhaus theorem about the conjugacy of complements is false in general. As a counterexample, consider the multiplicative group ℂ* of the complex number field ℂ and consider the p-Sylow for any prime p, or even the torsion part of ℂ*. Let H be this subgroup. H has a complement, but this complement is found by Zorn's Lemma (consider a maximal subgroup that intersects H trivially) and the use of Zorn's Lemma is essential. In fact, by Zorn's Lemma, any subgroup that has a trivial intersection with H can be extended to a complement of H. Since ℂ* is abelian, these complements cannot be conjugated to each other.

On any reasonable definition of functions, neither the category of sets nor the category of small categories is cartesian closed in New Foundations (NF). The latter category is sometimes proposed as a foundation for category theory since it is among its own objects. Our result shows it is a poor one.

In NF, as in other set theories, a "function" f from a set A to a set B is defined to be a set f of ordered pairs 〈x, y〉 with x in A and y in B, such that (a) if 〈x, y〉 ∈ f and 〈x, y′〉 ∈ f then y = y′, and (b) for every x in A there is some y in B with 〈x, y〉 ∈ f. But in NF different definitions of ordered pairs give significantly different functions. I say a reasonable definition must give:

1. The formula z = 〈x, y〉 is stratifiable.

2. For every set S there is a set {〈x, x〉 ∣ x ∈ S}.

3. If f is a function from A to B, and g one from B to C, there is a set {〈x, z〉∣(∃y)〈x, y〉∈ f & 〈y, z〉∈ g}.

Principles 2 and 3 are needed for identity functions and composites. By principle 1, any sets A and B have a set A × B of all ordered pairs 〈x, y〉 with x in A and y in B, but it does not follow that functions exist making A × B a categorical product of A and B.

By means of an Ehrenfeucht-Mostowski construction we obtain an automorphism theorem for a syntactically characterized class of Laa-theories comprising in particular the finitely determinate ones. Examples of Laa-theories with only rigid models show this result to be optimal with respect to a classification in terms of prenex quantifier type: Rigidity is seen to hinge on quantification of type … ∀ … stat … permitting of the parametrization of families of disjoint stationary systems by the elements of the universe.

The purpose of this paper is an axiomatic study of the interrelations between certain continuity properties. We deal with principles which are equivalent to the statements “every mapping is sequentially nondiscontinuous”, “every sequentially nondiscontinuous mapping is sequentially continuous”, and “every sequentially continuous mapping is continuous”. As corollaries, we show that every mapping of a complete separable space is continuous in constructive recursive mathematics (the Kreisel-Lacombe-Schoenfield-Tsejtin theorem) and in intuitionism.

Let be a countable saturated structure, and assume that D(v) is a strongly minimal formula (without parameter) such that is the algebraic closure of D(). We will prove the two following theorems:

Theorem 1. If G is a subgroup of Aut() of countable index, there exists a finite set A in such that every A-strong automorphism is in G.

Theorem 2. Assume that G is a normal subgroup of Aut() containing an element g such that for all n there exists X ⊆ D() such that Dim(g(X)/X) > n. Then every strong automorphism is in G.

Let L be a finite relational language. The age of a structure over L is the set of isomorphism types of finite substructures of . We classify those ages for which there are less than 2ω countably infinite pairwise nonisomorphic L-structures of age .

We consider some modal languages with a modal operator D whose semantics is based on the relation of inequality. Basic logical properties such as definability, expressive power and completeness are studied. Also, some connections with a number of other recent proposals to extend the standard modal language are pointed at.

An -structure is called internally presented in a nonstandard universe if its base set and interpretation of every symbol in are internal. A nonstandard universe is said to satisfy the κ-isomorphism property if for any two internally presented -structures and , where has less than κ many symbols, is elementarily equivalent to implies that is isomorphic to . In this paper we prove that the ℵ1-isomorphism property is equivalent to the ℵ0-isomorphism property plus ℵ1-saturation.

In this paper it is shown that the meet-inaccessible degrees are dense in R. The construction uses an 0′-priority argument. As a consequence, the meet-inaccessible degrees and the meet-accessible degrees give a partition of R into two sets, either of which is a nontrivial dense subset of R and generates R − {0} under joins (thus an automorphism base of R).

What is an inference rule? This question does not have a unique answer. One usually finds two distinct standard answers in the literature; validity inference (σ ⊦vφ for every substitution τ, the validity of τ[σ] entails the validity of τ[φ]), and truth inference (σ⊦l φ if for every substitution τ, the truth of τ[σ] entails the truth of τ[φ]). In this paper we introduce a general semantic framework that allows us to investigate the notion of inference more carefully. Validity inference and truth inference are in some sense the extremal points in our framework. We investigate the relationship between various types of inference in our general framework, and consider the complexity of deciding if an inference rule is sound, in the context of a number of logics of interest: classical propositional logic, a nonstandard propositional logic, various propositional modal logics, and first-order logic.

Under a sharp version of the assumption I∆0 ⊢ ¬ ∆0H we characterize models of I∆0 + BΣ1 having a proper end extension to a model of I∆0.

In [WP] Wilkie and Paris study the relationship between the existence of a proper end extension of a model and its “fullness”, which is related to a certain weak overspill principle (we recall the definition of fullness in §3). Let M be a countable nonstandard model of I∆0 + BΣ1. Under the hypothesis I∆0 ⊢ ¬ ∆0H they prove the following (Corollaries 7 and 8):

The following are equivalent:

1. 1) M has a proper end extension to a model of I∆0 + BΣ1.

2. 2) M is (I∆0 + BΣ1)-full.

Moreover, assuming that there is no t ∈ M such that for v ∈ M, 2[t/v]exists if and only if v < N, the following are equivalent:

1. 1) M has a proper end extension to a model of I∆0.

2. 2) M is I∆0-full.

Wilkie and Paris ask whether the assumption on the structure of the model can be eliminated from the second equivalence.

We eliminate it, but we sharpen the assumption I∆0 ⊢ ¬ ∆0H. So we partially answer their question.

The structure of the 1-degrees included in an m-degree with a maximal set together with the 1-reducibility relation is characterized. For this a special sublattice of the lattice of recursively enumerable sets under the set-inclusion is used.