The Scott height of a structure gives ordinal measure of the inhomogeneity of the structure. The Scott specturm of a collection of structures is the set of Scott heights of structures in the collection. We will recall the precise definitions of these and related concepts in the next section. The reader thoroughly unfamiliar with these notions may want to skip ahead before reading the rest of this Introduction.

In [11] it is shown that every model of the complete theory of (, +, 1), where, as usual, denotes the integers, is ℵ0-homogeneous, and therefore has Scott height at most ω. On the other hand, a footnote in [1] gives a model of the theory of (, +) which is not ℵ0-homogeneous, while in [11] such a model is described which can be expanded to a model of the theory of (, +, 1). However, since it is also true that any model of the theory of (, +) is isomorphic to a subgroup, in fact a pure subgroup, of a direct sum of and a torsion-free divisible group, it is easy to see that any such model must be ≡∞ω to a model of cardinality at most and so must have Scott height below .

After having recalled the relevant material about Scott heights in §2, we will review the situation for torsion abelian groups in §3. In §4 we shall produce torsion-free abelian groups of high Scott height. It is the proof of Theorem 15 that was our primary motivation in writing this paper.

We prove:

Theorem. A complete first order theory in a countable language which is strictly stable, trivial and which admits finite coding hasnonisomorphic countable models.

Combined with the corresponding result or superstable theories from [4] our result confirms the Vaught conjecture for trivial theories which admit finite coding.

We present a solution to the problem of denning a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by Lawvere-Tierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topos-theoretic results.

In this paper it is shown that the global statement that the dominating number for κ is less than 2κ for all regular κ, is internally consistent, given the existence of 0#. The possible range of values for the dominating number for κ and 2κ which may be simultaneously true in an inner model is also explored.

In [4] it is shown that only using exponentiation can one prove the existence of non trivial solutions of Pell equations in IΔ0. However, in this paper we will prove that any Pell equation has a non trivial solution modulo m for every m in IΔ0.

The problem of the nonequivalent modalities available in certain systems is a classical problem of modal logic. In this paper we deal with this problem without referring to particular logics, but considering the whole class of normal propositional logics. Given a logic L let P(L) (the m-partition generated by L) denote the set of the classes of L-equivalent modalities. Obviously, different logics may generate the same m-partition; the first problem arising from this general point of view is therefore to determine the cardinality of the set of all m-partitions. Since, as is well known, there exist normal logics, and since one immediately realizes that there are infinitely many m-partitions, the problem consists in choosing (assuming the continuum hypothesis) between ℵ0 and . In Theorem 1.2 we show that there are m-partitions, as many as the logics.

The next problem which naturally arises consists in determining, given an m-partition P(L), the number of logics generating P(L) (in symbols, μ(P(L))). In Theorem 2.1(ii) we show that ∣{P(L): μ(P(L)) = }∣ = . Now, the set {L′) = P(L)} has a natural minimal element; that is, the logic L* axiomatized by K ∪ {φ(p) ↔ ψ(p): φ, ψ are L-equivalent modalities}; P(L) and L* can be, in some sense, identified, thus making the set of m-partitions a subset of the set of logics.

This paper looks at the concept of neighborhood canonicity introduced by Brian Chellas [2]. We follow the lead of the author's paper [9] where it was shown that every non-iterative logic is neighborhood canonical and here we will show that all logics whose axioms have a simple syntactic form—no intensional operator is in boolean combination with a propositional letter—and which have the finite model property are neighborhood canonical. One consequence of this is that KMcK, the McKinsey logic, is neighborhood canonical, an interesting counterpoint to the results of Robert Goldblatt and Xiaoping Wang who showed, respectively, that KMcK is not relational canonical [5] and that KMcK is not relationally strongly complete [11].

We present a characterization of weakly compact cardinals in terms of generalized stationarity. We apply this characterization to construct a model with no partial square sequences.

We introduce the class PIO of functions computable in time that is polynomial in max {the length of input, the length of output}, observe that there is no notation system for total PIO functions but there are notation systems for partial PIO functions, and give an algebra of partial PIO functions from binary strings to binary strings.

For the reader's convenience, we begin Part II by restating the rules for constructing Sq derivations given in §2:

We let R be the set of 14 rules for Sq, R* the set of pure *-rules in R, R≃ the set of pure ≃-rules in R, and R− the set R ∩ − {Exs≃}. The closure of any F ⊆ Sq under R will be denoted by csR F, and the direct image of F under Cut*, for example, by Cut*(F).

We study the cardinal invariants of measure and category after adding one random real. In particular, we show that the number of measure zero subsets of the plane which are necessary to cover graphs of all continuous functions may be large while the covering for measure is small.