A strong partition cardinal is an uncountable well-ordered cardinal κ such that every partition of [κ]κ (the size κ subsets of κ) into less than κ many pieces has a homogeneous set of size κ. The existence of such cardinals is inconsistent with the axiom of choice, and our work concerning them is carried out in ZF set theory with just dependent choice (DC). The consistency of strong partition cardinals with this weaker theory remains an open question. The axiom of determinacy (AD) implies that a large number of cardinals including ℵ1 have the strong partition property. The hypothesis that AD holds in the inner model of constructible sets built over the real numbers as urelements has important consequences for descriptive set theory, and results concerning strong partition cardinals are often applied in this context. Kechris [4] and Kechris et al. [5] contain further information concerning the relationship between AD and strong partition cardinals.

We assume familiarity with the basic results on strong partition cardinals as developed in Kleinberg [6], [7], [8] and Henle [2]. Recall that a strong partition cardinal κ is measurable; in fact every stationary subset of κ is measure one under some normal measure on κ. If μ is a countably additive ultrafilter extending the closed unbounded filter on κ, then the length of the ultrapower [κ]κ under the less than almost everywhere μ ordering is again a measurable cardinal. In §1 below we establish a polarized partition property on these measurable cardinals.

In Baire space we define a sequence of equivalence relations ‹Ev ∣ v < , each Ev being with classes in + v + 1 and such that (i) Ev does not have perfectly many classes, and (ii) is countable iff < ω1. This construction can be extended cofinally in . A new proof is given of a theorem of Hausdorff on partitions of R into ω1 many sets.

One of the main ideas of calculi of natural deduction, as introduced by Jaśkowski and Gentzen, is that assumptions may be discharged in the course of a derivation. As regards sentential logic, this conception will be extended in so far as not only formulas but also rules may serve as assumptions which can be discharged. The resulting calculi and derivations with rules of any finite level are informally introduced in §1, while §§2 and 3 state formal definitions of the concepts involved and basic lemmata. Within this framework, a standard form for introduction and elimination rules for arbitrary n-ary sentential operators is motivated in §4, understood as a contribution to the theory of meaning for logical signs. §5 proves that the set {&, ∨, ⊃, ⋏} of standard intuitionistic connectives is complete, i.e. &, ∨, ⊃, and ⋏ suffice to express each n-ary sentential operator having rules of the standard form given in §4. §6 makes some remarks on related approaches. For an extension of the conception presented here to quantifier logic, see [11].

The jump hierarchy of Turing degrees assigns to each ξ < (ℵ1)L the degree 0(ξ); we presuppose familiarity with its definition and with the basic terminology of [5]. Let λ be a limit ordinal, λ < (ℵ1)L. The central result of [5] concerns the relation between 0(λ) and exact pairs on Iλ = {0(ξ) ∣ ξ < λ}. In [6] this question is raised: Where a is an upper bound on Iλ, how far apart are a and 0(λ)? It is there shown that if λ is locally countable and admissible, they may be very far apart: 0(λ) = the least member of {a(Ind(λ))∣, a is an upper bound on Iλ}; this is rather pathological, for Ind(λ) may be larger than λ. If λ is locally countable but neither admissible nor a limit of admissibles, we are essentially in the case of λ < ; by results of Sacks [12] and Enderton and Putnam [2], 0(λ) = the least member of {a(2) ∣ a is an upper bound on Iλ}. If λ is not locally countable, Ind(λ) is neither admissible nor a limit of admissibles, so we are again in a case like that of λ < . But what if λ is locally countable and nonadmissible, but is a limit of admissibles? For the rest of this paper let λ be such an ordinal. The central result of this paper answers this question for some such λ.

A type-structure of partial effective functionals over the natural numbers, based on a canonical enumeration of the partial recursive functions, is developed. These partial functionals, defined by a direct elementary technique, turn out to be the computable elements of the hereditary continuous partial objects; moreover, there is a commutative system of enumerations of any given type by any type below (relative numberings).

By this and by results in [1] and [2], the Kleene-Kreisel countable functionals and the hereditary effective operations (HEO) are easily characterized.

In [4] I proved that in any nontrivial algebraic language there are no algorithms which enable us to decide whether a given finite set of equations Σ has each of the following properties except P2 (for which the problem is open):

P0(Σ) = the equational theory of Σ is equationally complete.

P1(Σ) = the first-order theory of Σ is complete.

P2(Σ) = the first-order theory of Σ is model-complete.

P3(Σ) = the first-order theory of the infinite models of Σ is complete.

P4(Σ) = the first-order theory of the infinite models of Σ is model-complete.

P5(Σ) = Σ has the joint embedding property.

In this paper I prove that, in any finite trivial algebraic language, such algorithms exist for all the above Pi's. I make use of Ehrenfeucht's result [2]: The first-order theory generated by the logical axioms of any trivial algebraic language is decidable. The results proved here are part of my Ph.D. thesis [3]. I thank Wilfrid Hodges, who supervised it.

Throughout the paper is a finite trivial algebraic language, i.e. a first-order language with equality, with one operation symbol f of rank 1 and at most finitely many constant symbols.

Among the more traditional semantics for intuitionistic logic the Beth and the Kripke semantics seem well-suited for direct manipulations required for the derivation of metamathematical results. In particular Smoryński demonstrated the usefulness of Kripke models for the purpose of obtaining closure properties for first-order arithmetic, [S], and second-order arithmetic, [J-S]. Weinstein used similar techniques to handle intuitionistic analysis, [W]. Since, however, Beth-models seem to lend themselves better for dealing with analysis, cf. [D], we have developed a somewhat more liberal semantics, that shares the features of both Kripke and Beth semantics, in order to obtain analogues of Smoryński's collecting operations, which we will call Smoryński-glueing, in line with the categorical tradition.

An impressive theory has been developed, largely by Shelah, around the notion of a stable theory. This includes detailed structure theorems for the models of such theories as well as a generalized notion of independence. The various stability properties can be defined in terms of the numbers of types over sets, or in terms of the complexity of definable sets. In the concrete examples of stable theories, however, one finds an important distinction between “positive” and “negative” information, such a distinction not being an a priori consequence of the general definitions. In the naive examples this may take the form of distinguishing between say a class of a definable equivalence relation and the complement of a class. In the more algebraic examples, this distinction may have a “topological” significance, for example with the Zariski topology on (the set of n-tuples of) an algebraically closed field, the “closed” sets being those given by sets of polynomial equalities. Note that in the latter case, every definable set is a Boolean combination of such closed sets (the definable sets are precisely the constructible sets). Similarly, stability conditions in practice reduce to chain conditions on certain “special” definable sets (e.g. in modules, stable groups). The aim here is to develop and present such notions in the general (model-theoretic) context. The basic notion is that of an “equation”. Given a complete theory T in a language L, an L-formula φ(x̄, ȳ) is said to be an equation (in x̄) if any collection Φ of instances of φ(i.e. of formulae φ(x̄, ā)) is equivalent to a finite subset Φ′ ⊂ Φ.

In [1] Craig introduced a proof procedure for first order classical logic called linear reasoning. In it, a proof of P ⊃ Q consists of a sequence of formulas, each of which implies the next, beginning with P and ending with Q. And one of the formulas in the sequence will be an interpolation formula for P ⊃ Q. Indeed, this was the first proof of the Craig interpolation theorem, some of whose important consequences were demonstrated in a companion paper [2]. In this paper we present systems of linear reasoning for several standard modal logics: K, T, K4, S4, D, D4, and GL. Similar systems can be constructed for several regular, nonnormal modal logics too, though we do not do so here. And just as in the classical case, interpolation theorems are easy consequences. Such theorems are well known for the logics considered here. There is a model theoretic argument in [6], an argument using Gentzen systems in [8], an argument using consistency properties in [4] and [5], and an argument using symmetric Gentzen systems in [5]. This paper presents what seems to be the first modal proof that follows Craig's original methods. We note that if the modal rules given here are dropped, a classical linear reasoning system results, which is related to, but not the same as those in [1] and [10].

Since the basic linear reasoning ideas are fully illustrated by the propositional case, we present that first, to keep the clutter down. Later we show how the techniques can generally be extended to encompass quantifiers. We do not follow [1] in making heavy use of prenex form, since it is not generally available in modal logics. Fortunately, it plays no essential role.

In this paper we investigate the connections between categoricity and ranks. We use stability theory to prove some old and new results.

Much of the literature on the model theory of modal logics suffers from two weaknesses. Firstly, there is a lack of generality; theorems are proved piecemeal about this or that modal logic, or at best small classes of logics. Much of the literature, e.g. [1], [2], and [3], confines attention to structures with the expanding domain property (i.e., if wRu then Ā(w) ⊆ Ā(u)); the syntactic counterpart of this restriction is assumption of the converse Barcan scheme, a move which offers (in Russell's phrase) “all the advantages of theft over honest toil”. Secondly, I think there has been a failure to hit on the best ways of extending classical model theoretic notions to modal contexts. This weakness makes the literature boring, since a large part of the interest of modal model theory resides in the way in which classical model theoretic notions extend, and in some cases divide, in the modal setting. (The relation between α-recursion theory and classical recursion theory is analogous to that between modal model theory and classical model theory. Much of the work in α-recursion theory involved finding the right definitions (e.g., of recursive-in) and separating concepts which collapse in the classical case (e.g. of finiteness and boundedness).)

The notion of a well-behaved modal logic is introduced in §3 to make possible rather general results; of course our attention will not be restricted to structures with the expanding domain property. Rather than prove piecemeal that familiar modal logics are well-behaved, in §4 we shall consider a class of “special” modal logics, which obviously includes many familiar logics and which is included in the class of well-behaved modal logics.