We prove that certain syntactic conditions similar to separation principles on a theory are equivalent to semantic properties such as amalgamation and strong amalgamation, by showing that appropriate classes of structures are definable by Lω1ω-sentences. Then we characterise the elements of core models and thus give a natural proof of Rabin's characterisation of convex theories.

The notion of a syntactic characterisation of a semantic property of a theory is by now fairly well known. The earliest such were the classical preservation theorems: For example, a theorem of Lyndon characterised the theories whose models were closed under homomorphic images as those with a set of positive axioms.

Presumably the notion of syntactic characterisation can be made precise, but it is probably better at this stage to leave it vague. The general idea is that theories are “algebras” (cylindric algebras, or logical categories, with suitable extra structure) and that a semantic property P of theories is syntactically characterisable if the class of theories with P is an “elementary” class of “algebras.”

When one codes countable theories as real numbers, a syntactically characterisable property will be arithmetical. The converse does not seem reasonable, especially as it is often fairly easy to prove a property arithmetical (using extra predicates, usually), when we may not be able to find a syntactic characterisation.

For any linear order type τ with first element let be the Boolean set algebra generated by the left-closed right-open (including [x, ∞)) intervals of some linear order of type τ. Let η and ω be the order types of the rationals and natural numbers respectively (when not used as an order type, ω will, as usual, be the set of nonnegative integers). We show that the Boolean algebra of the elementary theory of well-orders, i.e., the Boolean algebra of elementary classes of well-orders or, equivalently, the algebra of equivalence classes of sentences of the elementary theory of well-orders, is isomorphic to and that the Boolean algebra of the elementary theory of abelian groups is isomorphic to . Both results are obtained by applying Hanf's structure diagram technique to the work of Mostowski, Tarski, and Szmielew. One may formalize discussion of algebras of proper classes by assuming the classes are included in a universe which is a set in a larger universe. Given a Boolean algebra, let 0 be its zero and 1 its unit, let ≤ be its associated partial ordering, and, for any elements a, b, and c of the algebra, let “a + b = c ” be the assertion that c is the disjoint sum of a and b. A subset of the algebra disjointly generates the algebra iff each of the algebra's elements is a disjoint sum of a finite number of the subset's elements.

We examine the action of unary functions on the regressive isols. A manageable theory is produced and we find that such a function maps ⋀R into ⋀ if and only if it is eventually R↑ increasing and maps ⋀R into ⋀R if and only if it is eventually recursive increasing. Our paper concludes with a discussion of other methods for extending functions to ⋀R.

The main purpose of this paper is to show that there exists a prime differentially closed extension over each differentially perfect field. We do this in a roundabout manner by first giving new and simple axioms for the theory of differentially closed fields (in the manner of Blum [1] for characteristic 0) and by proving that this theory is the model completion of the theory of differentially perfect fields. This paper can be read independently from [10], where we gave more complicated axioms for the same theory (in the manner of Robinson [6] for characteristic 0).

I am indebted to E. R. Kolchin for answering many questions and for making the manuscript of his forthcoming book [2] available to me.

A common method of obtaining the classical modal logics, for example the Feys system T, the Lewis systems, the Brouwerian system etc., is to build on a basis for the propositional calculus by adjoining a new symbol L, specifying new axioms involving L and the symbols in the basis for PC, and imposing one or more additional transformation rules. In the jargon of algebraic logic, which is the point of view we shall adopt, the “necessity” symbol L may be interpreted as an operator on the Boolean algebra of propositions of PC. For example, the Lewis system S4 may be regarded as a Boolean algebra ℒ together with an operator L on ℒ having the properties: (1)Lp ≤ p for all p in ℒ, (2)L1 = 1, (3) L(p → q) ≤ Lp → Lq for all p, q in ℒ, and (4) Lp = L(Lp) for all p in ℒ. Here, of course, → denotes the material implication connective: p→q = p′ ∨ q. It is easy to verify that property (3) may be replaced by either (3′) L(p ∧ q) = Lp ∧ Lq for all p, q in ℒ, or by (3″) L(p → q) ∧ Lp ≤ Lq for all p, q in ℒ. In particular, it follows from (1) through (4) above that L is a decreasing, idempotent and isotone operator on ℒ. Such mappings are often called interior operators.

In a previous paper [5], we considered the problem of introducing an implication connective into a quantum logic. This is greatly complicated by the fact that the quantal propositions band together to form an orthocomplemented lattice which is only “locally” distributive. Such lattices are called orthomodular. For definitions and further discussion, the reader is referred to that paper. In it, we argued that the Sasaki implication connective ⊃ defined by p ⊃ q = p′ ∨ (p ∧ q) is a natural generalization of material implication when the lattice of propositions is ortho-modular. Indeed, if unrestricted distributivity were permitted, p ⊃ q would reduce to the classical material implication p → q. For this reason, we choose ⊃ to play the role of material implication in an orthomodular lattice. Further properties of ⊃ are enumerated in Example 2.2(1) and Corollary 2.4 below.

In this paper we investigate some families of decision problems associated with a restricted class of Post canonical forms, specifically, those defined over one-letter alphabets whose productions have single premises and contain only one variable. For brevity sake, we call any such form an RPCF (Restricted Post Canonical Form). Constructive proofs are given which show, for any prescribed nonrecursive r.e. many-one degree of unsolvability D, the existence of an RPCF whose word problem is of degree D and an RPCF with axiom whose-decision problem is also of degree D. Finally, we show that both of these results are best possible in that they do not hold for one-one degrees.

This paper is concerned with extending some basic results from classical model theory to modal logic.

In §1, we define the majority of terms used in the paper, and explain our notation. A full catalogue would be excessive, and we cite [3] and [7] as general references.

Many papers on modal logic that have appeared are concerned with (i) introducing a new modal logic, and (ii) proving a weak completeness theorem for it. Theorem 1, in §2, in many cases allows us to conclude immediately that a strong completeness theorem holds for such a logic in languages of arbitrary cardinality. In particular, this is true of S4 with the Barcan formula.

In §3 we strengthen Theorem 1 for a number of modal logics to deal with the satisfaction of several sets of sentences, and so obtain a realizing types theorem. Finally, an omitting types theorem, generalizing the result for classical logic (see [5]) is proved in §4.

Several consequences of Theorem 1 are already to be found in the literature. [2] gives a proof of strong completeness in languages of arbitrary cardinality of various logics without the Barcan formula, and [8] for some logics in countable languages with it. In the latter case, the result for uncountable languages is cited, without proof, in [1], and there credited to Montague. Our proof was found independently.

In this paper we show the finite controllability of the Maslov class of formulas of pure quantification theory (specified immediately below). That is, we show that every formula in the class has a finite model if it has a model at all. A signed atomic formula is an atomic formula or the negation of one; a binary disjunction is a disjunction of the form A1 ⋁ A2, where A1 and A2 are signed atomic formulas; and a formula is Krom if it is a conjunction of binary disjunctions. Finally, a prenex formula is Maslov if its prefix is ∃···∃∀···∀∃···∃ and its matrix is Krom.

A number of decidability results have been obtained for formulas classified along these lines. It is a consequence of Theorems 1.7 and 2.5 of [4] that the following are reduction classes (for satisfiability): the class of Skolem formulas, that is, prenex formulas with prefixes ∀···∀∃···∃, whose matrices are conjunctions one conjunct of which is a ternary disjunction and the rest of which are binary disjunctions; and the class of Skolem formulas containing identity whose matrices are Krom. Moreover, the following results (for pure quantification theory, that is, without identity) are derived in [1] and [2]: the classes of prenex formulas with Krom matrices and prefixes ∃∀∃∀, or prefixes ∀∃∃∀, or prefixes ∀∃∀∀ are all reduction classes, while formulas with Krom matrices and prefixes ∀∃∀ comprise a decidable class. The latter class, however, is not finitely controllable, for it contains formulas satisfiable only over infinite universes. The Maslov class was shown decidable by Maslov in [11].

Let Q be the class of closed quantificational formulas ∀x∃u∀yM without identity such that M is a quantifier-free matrix containing only monadic and dyadic predicate letters and containing no atomic subformula of the form Pyx or Puy for any predicate letter P. In [DKW] Dreben, Kahr, and Wang conjectured that Q is a solvable class for satisfiability and indeed contains no formula having only infinite models. As evidence for this conjecture they noted the solvability of the subclass of Q consisting of those formulas whose atomic subformulas are of only the two forms Pxy, Pyu and the fact that each such formula that has a model has a finite model. Furthermore, it seemed likely that the techniques used to show this subclass solvable could be extended to show the solvability of the full class Q, while the syntax of Q is so restricted that it seemed impossible to express in formulas of Q any unsolvable problem known at that time.

In 1966 Aanderaa refuted this conjecture. He first constructed a very complex formula in Q having an infinite model but no finite model, and then, by an extremely intricate argument, showed that Q (in fact, the subclass Q2 defined below) is unsolvable ([Aa1], [Aa2]). In this paper we develop stronger tools in order to simplify and extend the results of [Aa2]. Specifically, we show the unsolvability of an apparently new combinatorial problem, which we shall call the linear sampling problem (defined in §1.2 and §2.3). From the unsolvability of this problem there follows the unsolvability of two proper subclasses of Q, which we now define. For each i ≥ 0, let Pi be a dyadic predicate letter and let Ri be a monadic predicate letter.

It will be shown that propositional tense logic (with the Kripke relational semantics) may be regarded as a fragment of propositional modal logic (again with the Kripke semantics). This paper deals only with model theory. The interpretation of formal systems of tense logic as formal systems of modal logic will be discussed in [6].

The languages M and T, of modal and tense logic respectively, each have a countable infinity of propositional variables and the Boolean connectives; in addition, M has the unary operator ⋄ and T has the unary operators F and P. A structure is a pair <W, ≺<, where W is a nonempty set and ≺ is a binary relation on W. An assignment V assigns to each propositional variable p a subset V(p) of W. Then V(α)£ W is defined for all formulas α of M or T by induction:

We say that α is valid in <W, ≺>, or <W, ≺>, ⊨ α, if ⊩ (α) = W for every assignment V for <W, ≺>. If Γ is a set of formulas of M [T] and α is a formula of M [T], then α is a logical consequence of Γ, or Γ ⊩ α,if α is valid in every model of Γ i.e., in every structure in which all γ ∈ Γ are valid.

§1. Let α be an admissible ordinal which is also a limit of admissible ordinals (e.g. take any α such that α = α*, its projectum [5]). For any admissible γ ≦ α, let [γ) denote the initial segment of ordinals less than γ. A very general question that one might ask is the following: What conditions should one put on γ so that a certain statement true in Lα is ‘reflected’ to be true in Lγ? We cite some examples: (a) If β < γ < α, then Lα ⊨ “β cardinal” is ‘reflected’ to Lγ ⊨ “ β cardinal” (⊨ is just the satisfaction relation). (b) If β < γ < α and γ is a cardinal in Lα (called α-cardinal for short), then Lα ⊨ “β is not a cardinal” is ‘reflected’ to Lγ ⊨ “β is not a cardinal” This fact is used in Gödel's proof that V = L implies the Generalized Continuum Hypothesis. Our objective in this paper is to study a ‘reflection’ property of the following sort: Let A ⊆ [α) be an α-recursively enumerable (α-r.e.), non-α-recursive set. Under what conditions will A restricted to a smaller admissible ordinal γ be γ-r.e. and not γ-recursive?

The notations used here are standard. Those that are not explained are adopted from the paper of Sacks and Simpson [5], to which we also refer the reader for background material.

Some of the theorems contained in the interesting and incisive paper [*] (D. J. Shoesmith and T. J. Smiley, Deducibility and many-valuedness, this Journal, vol. 36 (1970), pp. 610–622) are strongly connected with some earlier results, especially those presented in the paper of Łoś and Suszko [2]. The lack of appropriate bibliographical references in [*] shows that some results which have been obtained relatively long ago are, unfortunately, not so widely known as one might expect. The main task of this short note is to inform briefly on a few results and problems related to those examined in [*].

The Kondo-Addison theorem says that every set A of reals contains a real which is implicit (for short, Imp). Any real which is Imp is constructible, and it is the main purpose of this note to show that a close examination of the usual Kondo-Addison argument allows us to compute (in terms of the ordinals associated with A by the classical representation theorem) where in the constructible hierarchy an element of A ∩ Imp must occur. This result is related to a basis theorem of Barwise and Fisher [1] and is accompanied by a strong counterexample which shows that certain tepid improvements of their theorem (and, a fortiori, of the main theorem in this note) are impossible. As a final flourish, the ‘classifying ordinals’ used to compute the locations of the singletons are themselves characterized using an idea from [1].

The least possible jump for a degree of unsolvability a is its join a ∪ 0′ with 0′. Friedberg [1] showed that each degree b ≥ 0′ is the jump of a degree a realizing least possible jump (i.e., satisfying the equation a′ = a ∪ 0′). Sacks (cf. Stillwell [8]) showed that most (in the sense of Lebesgue measure) degrees realize least possible jump. Nevertheless, degrees not realizing least possible jump are easily found (e.g., any degree b ≥ 0′) even among the degrees <0′ (cf. Shoenfield [5]) and the recursively enumerable (r.e.) degrees (cf. Sacks [3]).

A degree is called minimal if it is minimal in the natural partial ordering of degrees excluding least element 0. The existence of minimal degrees <0” was first shown by Spector [7]; Sacks [3] succeeded in replacing 0” by 0′ using a priority argument. Yates [9] asked whether all minimal degrees <0′ realize least possible jump after showing that some do by exhibiting minimal degrees below each r.e. degree. Cooper [2] subsequently showed that each degree b > 0′ is the jump of a minimal degree which, as corollary to his method of proof, realizes least possible jump. We show with the aid of a simple combinatorial device applied to a minimal degree construction in the manner of Spector [7] that not all minimal degrees realize least possible jump. We have observed in conjunction with S. B. Cooper and R. Epstein that the new combinatorial device may also be applied to minimal degree constructions in the manner of Sacks [3], Shoenfield [6] or [4] in order to construct minimal degrees <0′ not realizing least possible jump. This answers Yates' question in the negative. Yates [10], however, has been able to draw this as an immediate corollary of the weaker result by carrying out the proof in his new system of prioric games.

The Hanf number for sentences of a language L is defined to be the least cardinal κ with the property that for any sentence φ of L, if φ has a model of power ≥ κ then φ has models of arbitrarily large cardinality. We shall be interested in the language Lω1,ω (see [3]), which is obtained by adding to the formation rules for first-order logic the rule that the conjunction of countably many formulas is also a formula.

Lopez-Escobar proved [4] that the Hanf number for sentences of Lω1,ω is ⊐ω1, where the cardinals ⊐α are defined recursively by ⊐0 = ℵ0 and ⊐α = Σ{2⊐β: β < α} for all cardinals α > 0. Here ω1 denotes the least uncountable ordinal.

A sentence of Lω1,ω is complete if all its models satisfy the same Lω1,ω-sentences. In [5], Malitz proved that the Hanf number for complete sentences of Lω1,ω is also ⊐ω1, but his proof required the generalized continuum hypothesis (GCH). The purpose of this paper is to give a proof that does not require GCH.

More precisely, we will prove the following:

Theorem 1. For any countable ordinal α, there is a complete Lω1,ω-sentence σαwhich has models of power ⊐α but no models of higher cardinality.

Our basic approach is identical with Malitz's. We simply use a different combinatorial fact at the crucial point.

Suppose M is a countable standard transitive model of set theory. P. J. Cohen [2] showed that if κ is an infinite cardinal of M then there is a one-to-one function Fκ from κ into the set of real numbers such that M[Fκ] is a model of set theory with the same cardinals as M.

If Tκ is the range of Fκ then Cohen also showed [2] that M[Tκ] fails to satisfy the axiom of choice. We will give an easy proof of this fact.

If κ, λ are infinite we will also show that M[Tκ] is elementarily equivalent to M[Tλ] and that (] in M[Fλ]) is elementarily equivalent to (] in M[FK]).

Finally we show that there may be an N ∈ M[GK] which is a standard model of set theory (without the axiom of choice) and which has, from the viewpoint of M[GK], more real numbers than ordinals.

We write ZFC and ZF for Zermelo-Fraenkel set theory, respectively with and without the axiom of choice (AC). GBC is Gödel-Bernays' set theory with AC. DC and ACℵo are respectively the axioms of dependent choice and of countable choice defined in [6].

Lower case Greek characters (other than ω) are used as variables over ordinals. When α is an ordinal, R(α) is the set of all sets with rank less than α.

The principal aim of this paper is to establish a theorem stating, roughly, that the addition of the fan theorem and the. continuity schema to an intuitionistic system of elementary analysis results in a conservative extension with respect to arithmetical statements.

The result implies that the consistency of first order arithmetic cannot be proved by use of the fan theorem, in addition to standard elementary methods—although it was the opposite assumption which led Gentzen to withdraw the first version of his consistency proof for arithmetic (see [B]).

We must presuppose acquaintance with notation and principal results of [K, T], and with §1.6, Chapter II, and Chapter III, §4-6 of [T1]. In one respect we shall deviate from the notation in [K, T]: We shall use (n)x (instead of g(n, x)) to indicate the xth component of the sequence coded by n, if x < lth(n), 0 otherwise.

We also introduce abbreviations n ≤* m, a ≤ b which will be used frequently below: