Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators or, in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. The two direct translations turn out to be complete. The paper fulfills the program of Church [1932], [1933] and Curry [1930] to base logic on a consistent system of λ-terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent).

A minimal formula is a formula which is minimal in provable formulas with respect to the substitution relation. This paper shows the following: (1) A β-normal proof of a minimal formula of depth 2 is unique in NJ. (2) There exists a minimal formula of depth 3 whose βη-normal proof is not unique in NJ. (3) There exists a minimal formula of depth 3 whose βη-normal proof is not unique in NK.

The interest in fragments of predicate logics is motivated by the well-known fact that full classical predicate calculus is undecidable (cf. Church [1936]). So it is desirable to find decidable fragments which are in some sense “maximal”, i.e., which become undecidable if they are “slightly” extended. Or, alternatively, we can look for “minimal” undecidable fragments and try to identify the vague boundary between decidability and undecidability. A great deal of work in this area concerning mainly classical logic has been done since the thirties. We will not give a complete review of decidability and undecidability results in classical logic, referring the reader to existing monographs (cf. Suranyi [1959], Lewis [1979], and Dreben, Goldfarb [1979]). A short summary can also be found in the well-known book Church [1956]. Let us recall only several facts. Herein we will consider only logics without functional symbols, constants, and equality.

(C1) The fragment of the classical logic with only monadic predicate letters is decidable (cf. Behmann [1922]).

(C2) The fragment of the classical logic with a single binary predicate letter is undecidable. (This is a consequence of Gödel [1933].)

(C3) The fragment of the classical logic with a single individual variable is decidable; in fact it is equivalent to Lewis S5 (cf. Wajsberg [1933]).

(C4) The fragment of the classical logic with two individual variables is decidable (Segerberg [1973] contains a proof using modal logic; Scott [1962] and Mortimer [1975] give traditional proofs.)

(C5) The fragment of the classical logic with three individual variables and binary predicate letters is undecidable (cf. Surańyi [1943]). In fact this paper considers formulas of the following type

φ,ψ being quantifier-free and the set of binary predicate letters which can appear in φ or ψ being fixed and finite.

In recent work, Cooper [3, 1990] has extended results of Jockusch and Shore [6, 1984] to show that the Turing jump is definable in the structure given by the Turing degrees and the ordering of Turing reducibility. In his definition of x′ from x, Cooper identifies an order-theoretic property shared by all of the degrees that are recursively enumerable in x and above x. He then shows that x′ is the least upper bound of all the degrees with this property. Thus, the jump of x is identified by comparing the recursively enumerable degrees with other degrees which are not recursively enumerable. Of course, once the jump operator is known to be definable, the relation of jump equivalence x′ = y′ is also known to be a definable relation on x and y. If we consider how much of the global theory of the Turing degrees is sufficient for Cooper's methods, it is immediately clear that his methods can be implemented to show that the jump operator and its weakening to the relation of jump equivalence are definable in any ideal closed under the Turing jump. However, his methods do not localize to , the degrees, or to the recursively enumerable degrees.

This paper fits, as do Shore and Slaman [16, 1990] and [17, to appear], within the general project to develop an understanding of the relationship between the local degree-theoretic properties of a recursively enumerable set A and its jump class. For an analysis of the possibility of defining jump equivalence in , consult Shore [15, to appear] who shows that the relation x(3) = y(3) is definable. In this paper, we will restrict our attention to definitions expressed completely in ℛ (Note: All sets and degrees discussed for the remainder of this paper will be recursively enumerable.) Ultimately, one would like to find some degree-theoretic properties definable in terms of the ordering of Turing reducibility and quantifiers over the recursively enumerable degrees that would define the relation of jump equivalence or define one or more of the jump classes Hn = {w∣ wn = 0n+1} or Ln = {w ∣ wn = 0n}. Such a result could very likely then be used as a springboard to other general definability results for the recursively enumerable degrees. It would be especially interesting to know whether every recursively enumerable degree is definable and whether every arithmetical degree-invariant property of the recursively enumerable sets is definable in .

We will consider some partition properties of the following type: given a function F: ωω →2, is there a sequence H0, H1, … of subsets of ω such that F is constant on ΠiεωHi? The answer is obviously positive if we allow all the Hi's to have exactly one element, but the problem is nontrivial if we require the Hi's to have at least two elements. The axiom of choice contradicts the statement “for all F: ωω→ 2 there is a sequence H0, H1, H2,… of subsets of ω such that {i|(Hi) ≥ 2} is infinite and F is constant on ΠHi”, but the infinite exponent partition relation ω(ω)ω implies it; so, this statement is relatively consistent with an inaccessible cardinal. (See [1] where these partition properties were considered.)

We will also consider partitions into any finite number of pieces, and we will prove some facts about partitions into ω-many pieces.

Given a partition F: ωω → k, we say that H0, H1…, a sequence of subsets of ω, is homogeneous for F if F is constant on ΠHi. We say the sequence H0, H1,… is nonoverlapping if, for all i ∈ ω, ∪Hi > ∩Hi+1.

The sequence 〈Hi: i ∈ ω〉 is of type 〈α0, α1,…〉 if, for every i ∈ ω, ∣Hi∣ = αi.

We will adopt the usual notation for polarized partition relations due to Erdös, Hajnal, and Rado.

means that for every partition F: κ1 × κ2 × … × κn→δ there is a sequence H0, H1,…, Hn such that Hi ⊂ κi and ∣Hi∣ = αi for every i, 1 ≤ i ≤ n, and F is constant on H1 × H2 × … × Hn.

In this paper we extend the models for the “logic of categories” to a wider class of categories than is usually considered. We consider two kinds of logic, a restricted first-order logic and the full higher-order logic of elementary topoi.

The restricted first-order logic has as its only logical symbols ∧, ∃, Τ, and =. We interpret this logic in a category with finite limits equipped with a factorization system (in the sense of [4]). We require to satisfy two additional conditions: ⊆ Monos, and any pullback of an arrow in is again in . A category with a factorization system satisfying these conditions will be called an EM-category.

The interpretation of the restricted logic in EM-categories is given in §1. In §2 we give an axiomatization for the logic, and in §§3 and 5 we give two completeness proofs for this axiomatization. The first completeness proof constructs an EM-category out of the logic, in the spirit of Makkai and Reyes [8], though the construction used here differs from theirs. The second uses Boolean-valued models and shows that the restricted logic is exactly the ∧, ∃-fragment of classical first-order logic (adapted to categories). Some examples of EM-categories are given in §4.

The restricted logic is powerful enough to handle relations, and in §6 we assign to each EM-category a bicategory of relations Rel() and a category of “functional relations” fr. fr is shown to be a regular category, and it turns out that Rel( and Rel(fr) are biequivalent bicategories. In §7 we study complete objects in an EM-category where an object of is called complete if every functional relation into is yielded by a unique morphism into . We write c for the full subcategory of consisting of the complete objects. Complete objects have some, but not all, of the properties that sheaves have in a category of presheaves.

Let X be a standard Borel space (i.e., a Polish space with the associated Borel structure), and let E be a countable Borel equivalence relation on X, i.e., a Borel equivalence relation E for which every equivalence class [x]E is countable. By a result of Feldman-Moore [FM], E is induced by the orbits of a Borel action of a countable group G on X.

The structure of general countable Borel equivalence relations is very little understood. However, a lot is known for the particularly important subclass consisting of hyperfinite relations. A countable Borel equivalence relation is called hyperfinite if it is induced by a Borel ℤ-action, i.e., by the orbits of a single Borel automorphism. Such relations are studied and classified in [DJK] (see also the references contained therein). It is shown in Ornstein-Weiss [OW] and Connes-Feldman-Weiss [CFW] that for every Borel equivalence relation E induced by a Borel action of a countable amenable group G on X and for every (Borel) probability measure μ on X, there is a Borel invariant set Y ⊆ X with μ(Y) = 1 such that E ↾ Y (= the restriction of E to Y) is hyperfinite. (Recall that a countable group G is amenable if it carries a finitely additive translation invariant probability measure defined on all its subsets.) Motivated by this result, Weiss [W2] raised the question of whether every E induced by a Borel action of a countable amenable group is hyperfinite. Later on Weiss (personal communication) showed that this is true for G = ℤn. However, the problem is still open even for abelian G. Our main purpose here is to provide a weaker affirmative answer for general amenable G (and more—see below). We need a definition first. Given two standard Borel spaces X, Y, a universally measurable isomorphism between X and Y is a bijection ƒ: X → Y such that both ƒ, ƒ-1 are universally measurable. (As usual, a map g : Z → W, with Z and W standard Borel spaces, is called universally measurable if it is μ-measurable for every probability measure μ on Z.) Notice now that to assert that a countable Borel equivalence relation on X is hyperfinite is trivially equivalent to saying that there is a standard Borel space Y and a hyperfinite Borel equivalence relation F on Y, which is Borel isomorphic to E, i.e., there is a Borel bijection ƒ: X → Y with xEy ⇔ ƒ(x)F ƒ(y). We have the following theorem.

A Skolem class is a class of formulas of pure quantification theory in Skolem normal form: closed, prenex formulas with prefixes ∀…∀∃…∃. (Pure quantification theory contains quantifiers, truth-functions, and predicate letters, but neither the identity sign nor function letters.) The Gödel Class, in which the number of universal quantifiers is limited to two, was shown effectively solvable (for satisfiability) sixty years ago [G1]. Various solvable Skolem classes that extend the Gödel Class can be obtained by allowing more universal quantifiers but restricting the combinations of variables that may occur together in atomic subformulas [DG, Chapter 2]. The Gödel Class and these extensions are also finitely controllable, that is, every satisfiable formula in them has a finite model. In this paper we isolate a model-theoretic property that connects the usual solvability proofs for these classes and their finite controllability. For formulas in the solvable Skolem classes, the property is necessary and sufficient for satisfiability. The solvability proofs implicitly relied on this fact. Moreover, for any formula in Skolem normal form, the property implies the existence of a finite model.

The proof of the latter implication uses the random models technique introduced in [GS] for the Gödel Class and modified and applied in [Go] to the Maslov Class. The proof thus substantiates the claim made in [Go] that random models can be adapted to the Skolem classes of [DG, Chapter 2]. As a whole, the results of this paper provide a more general, systematic approach to finite controllability than previous methods.

In well-known papers ([A-K1], [A-K2], and [E]) J. Ax, S. Kochen, and J. Ershov prove a transfer theorem for henselian valued fields. Here we prove an analogue for henselian valued and ordered fields. The orders for which this result apply are the usual orders and also the higher level orders introduced by E. Becker in [Bl] and [B2]. With certain restrictions, two henselian valued and ordered fields are elementarily equivalent if and only if their value groups (with a little bit more structure) and their residually ordered residue fields (a henselian valued and ordered field induces in a natural way an order in its residue field) are elementarily equivalent. Similar results are proved for elementary embeddings and ∀-extensions (extensions where the structure is existentially closed).

Let E ⊲ F iff E and F are extenders and E ∈ Ult(V, F). Intuitively, E ⊲ F implies that E is weaker—embodies less reflection—than F. The relation ⊲ was first considered by W. Mitchell in [M74], where it arises naturally in connection with inner models and coherent sequences. Mitchell showed in [M74] that the restriction of ⊲ to normal ultrafilters is well-founded.

The relation ⊲ is now known as the Mitchell order, although it is not actually an order. It is irreflexive, and its restriction to normal ultrafilters is transitive, but under mild large cardinal hypotheses, it is not transitive on all extenders. Here is a counterexample. Let κ be (λ + 2)-strong, where λ > κ and λ is measurable. Let E be an extender with critical point κ and let U be a normal ultrafilter with critical point λ such that U ∈ Ult(V, E). Let i: V → Ult(V, U) be the canonical embedding. Then i(E) ⊲ U and U ⊲ E, but by 3.11 of [MS2], it is not the case that i(E) ⊲ E. (The referee pointed out the following elementary proof of this fact. Notice that i ↾ Vλ+2 ∈ Ult(V, E) and X ∈ Ea ⇔ X ∈ i(E)i(a). Moreover, we may assume without loss of generality that = support(E). Thus, if i(E) ∈ Ult(V, E), then E ∈ Ult(V, E), a contradiction.)

By going to much stronger extenders, one can show the Mitchell order is not well-founded. The following example is well known. Let j: V → M be elementary, with Vλ ⊆ M for λ = joω(crit(j)). (By Kunen, Vλ+1 ∉ M.) Let E0 be the (crit(j), λ) extender derived from j, and let En+1 = i(En), where i: V → Ult(V, En) is the canonical embedding. One can show inductively that En is an extender over V, and thereby, that En+1 ⊲ En for all n < ω. (There is a little work in showing that Ult(V, En+1) is well-founded.)

We investigate the connection between -stability for random and Cohen forcing notions and the measurability and categoricity of the -sets. We show that Shelah's model for -measurability and categoricity satisfies -random-stability while it does not satisfy -Cohen-stability. This gives an example of measure-category asymmetry. We also present a result concerning finite support iterations of Suslin forcing.

We consider reducts of the structure ℛ = 〈ℝ, +, ·, <〉 and other real closed fields. We compete the proof that there exists a unique reduct between 〈ℝ, +, <,λa〉a ∈ ℝ and ℛ, and we demonstrate how to recover the definition of multiplication in more general contexts than the semialgebraic one. We then conclude a similar result for reducts between 〈ℝ, ·, <〉 and ℛ and for general real closed fields.

‘How can we recognize, given axioms and inference rules of a calculus, whether the calculus has such-and-such property?’ A question of this kind arises whenever we deal with a new logic system. For large families of logics, this question may be considered as an algorithmic problem, and a property is called decidable in a given family if there exists an algorithm which is capable of deciding, for a finite axiomatics of a calculus in the family, whether or not it has the property.

In the class of intermediate propositional logics, for instance, nontrivial properties such as the tabularity, pretabularity, and interpolation property (Maksimova [1972, 1977]) are decidable. However, for many other important properties—decidability, finite model property, disjunction property, Halldén-completeness, etc.—effective criteria were not found in spite of considerable efforts.

In this paper we show that the difficulties in investigating these properties in the classes of intermediate logics and normal modal logics containing S4 are of principal nature, since all of them turn out to be algorithmically undecidable. In other words, there are no algorithms which, given a finite set of axioms of an intermediate or modal calculus, can recognize whether or not it is decidable, Halldén-complete, has the finite model or disjunction property.

The first results concerning the undecidability of properties of calculi seem to have been obtained by Linial and Post [1949], who proved the undecidability of the problem of equivalence to classical calculus in the class of all propositional calculi with the same language as the classical one and the two inference rules: modus ponens and substitution. Kuznetsov [1963] generalized this result having proved the undecidability of the problem of equivalence to any fixed intermediate calculus (for instance, to intuitionistic calculus or even the inconsistent one). However, these results will not hold if we confine ourselves only to the class of intermediate logics, though the problem of equivalence to the undecidable intermediate calculus of Shehtman [1978] is clearly undecidable in this class as well.

We discuss a ‘negative’ way of defining frame classes in (multi)modal logic, and address the question of whether these classes can be axiomatized by derivation rules, the ‘non-ξ rules’, styled after Gabbay's Irreflexivity Rule. The main result of this paper is a metatheorem on completeness, of the following kind: If ⋀ is a derivation system having a set of axioms that are special Sahlqvist formulas and ⋀+ is the extension of ⋀ with a set of non-ξ rules, then ⋀+ is strongly sound and complete with respect to the class of frames determined by the axioms and the rules.

We consider an infinite two-person game. The second player has a winning perfect information strategy; we show that this player has a winning strategy which depends on substantially less information. The game studied here is a variation on a game of Gale.

We generalise various properties of quasiendomorphisms from groups with regular generic to small abelian groups. In particular, for a small stable abelian group such that no infinite definable quotient is connected-by-finite, the ring of quasi-endomorphisms is locally finite. Under some additional assumptions, it decomposes modulo some nil ideal into a sum of finitely many matrix rings.

We study descriptive set theory in the space by letting trees with no uncountable branches play a similar role as countable ordinals in traditional descriptive set theory. By using such trees, we get, for example, a covering property for the class of -sets of .

We call a family of trees universal for a class of trees if ⊆ and every tree in can be order-preservingly mapped into a tree in . It is well known that the class of countable trees with no infinite branches has a universal family of size ℵ1. We shall study the smallest cardinality of a universal family for the class of trees of cardinality ≤ ℵ1 with no uncountable branches. We prove that this cardinality can be 1 (under ¬CH) and any regular cardinal κ which satisfies (under CH). This bears immediately on the covering property of the -subsets of the space .

We also study the possible cardinalities of definable subsets of . We show that the statement that every definable subset of has cardinality <ωn or cardinality is equiconsistent with ZFC (if n ≥ 3) and with ZFC plus an inaccessible (if n = 2).

Finally, we define an analogue of the notion of a Borel set for the space and prove a Souslin-Kleene type theorem for this notion.