In this paper we show that any countable upper semilattice with zero can be embedded as an initial segment of the degrees of unsolvability. This provides a characterization of the order types of the countable initial segments of the degrees since any such initial segment is trivially an initial segment of a countable upper semilattice. Let a segment S of the degrees be a set of degrees such that, if a, b Є S and a < c < b, then c Є S. One may readily observe that our result also characterizes the order types of all countable segments of the degrees of unsolvability.

Modal reduction principles (MRPs) are modal formulas of the following form: Mp → Np, where M, N are (possibly empty) sequences of modal operators (i.e. □ or ◊). The notation M, N will be used to abbreviate such an MRP. We study a certain semantic correspondence between modal formulas and relational properties and obtain two main results. (1) On transitive semantic structures every MRP corresponds to a first-order relational property. (2) For the general case a syntactic criterion exists for distinguishing modal formulas with corresponding first-order properties from the others.

Theorem. Assume that there exists a standard model of ZFC + V = L. Then there is a model of ZFC in which the partial ordering of the degrees of constructibility of reals is isomorphic with a given finite lattice.

The proof of the theorem uses forcing. The definition of the forcing conditions and the proofs of some of the lemmas are connected with Lerman's paper on initial segments of the upper semilattice of the Turing degrees [2]. As an auxiliary notion we shall introduce the notion of a sequential representation of a lattice, which slightly differs from Lerman's representation.

Let K be a given finite lattice. Assume that the universe of K is an integer l. Let ≤ K be the ordering in K. A sequential representation of K is a sequence Ui ⊆ Ui+1 of finite subsets of ω i such that the following holds:

(1) For any s, s′ Є Ui , i Є ω, k, m Є l, k ≤ K m & s(m) = s′(m) → s(k) = s′(k).

(2) For any s Є Ui , i Є ω, s(0) = 0 where 0 is the least element of K.

(3) For any s, s′ Є i Є ω, k,j Є l, if k y Kj = m and s(k) = s′(k) & s(j) = s′(j) → s(m) = s′(m), where v K denotes the join in K.

We provide a semantics for Zermelo-Fraenkel set theory. The semantics was essentially given by W. Ackermann [1], and the proof of completeness follows from a lemma of Levy [4]. Furthermore, a close relationship between Ackermann's set theory and Zermelo-Fraenkel set theory has been established by Levy [4] and Reinhardt [6]. Thus, any interest in this paper must consist solely in the explicit formulation of the completeness result and the consequential evidence of the fundamental character of Ackermann's principle and the ad hoc character of the full law of excluded middle and the strong transitivity of Ackermann's unary relation

In [B1] and [B2] we have shown that the classical theorems on the continuity of effective operations are not derivable in various constructive formal systems. The purpose of this paper is to establish that these systems are closed under the corresponding rules of inference, which have the form: If e can be proved to be (the index of) an effective operation, then e can be proved to be continuous. In fact we establish closure under apparently stronger rules of the form, if e can be proved to be an effective operation, then e* can be proved to be (the index of) a modulus of continuity for e, where e* is canonically obtained from e.

We again deal with two kinds of effective operations: (i) partial operations on partial arguments, and (ii) total operations on total arguments. The continuity theorem for operations of kind (i) is due to Myhill-Shepherdson [MS] and is called MS; for kind (ii) is due to Kreisel, Lacombe, and Shoenfield [KLS] and is called KLS. We also deal with effective operations on the (recursive) real numbers, which are of kind (ii). In [B1], [B2] it is shown that KLS and MS are derivable in in-tuitionistic arithmetic HA from Markov's principle MP (this follows from examination of the classical proofs); that they do not imply MP in HA; and that they are not derivable in HA. (These theorems remain true when HA is replaced by various stronger systems.) The derived rules to be established in this paper therefore represent nontrivial closure properties of the systems to which they apply.

This note describes a simple interpretation * of modal first-order languages K with but finitely many predicates in derived classical second-order languages L(K) such that if Γ is a set of K-formulae, Γ is satisfiable (according to Kripke's 55 semantics) iff Γ* is satisfiable (according to standard (or nonstandard) second-order semantics).

The motivation for the interpretation is roughly as follows. Consider the “true” modal semantics, in which the relative possibility relation is universal. Here the necessity operator can be considered a universal quantifier over possible worlds. A possible world itself can be identified with an assignment of extensions to the predicates and of a range to the quantifiers; if the quantifiers are first relativized to an existence predicate, a possible world becomes simply an assignment of extensions to the predicates. Thus the necessity operator can be taken to be a universal quantifier over a class of assignments of extensions to the predicates. So if these predicates are regarded as naming functions from extensions to extensions, the necessity operator can be taken as a string of universal quantifiers over extensions.

The alphabet of a “finite” modal first-order language K shall consist of a non-empty countable set Var of individual variables, a nonempty finite set Pred of predicates, the logical symbols ‘¬’ ‘∧’, and ‘∧’, and the operator ‘◊’. The formation rules of K generate the usual Polish notations as K-formulae. ‘ν’, ‘ν1’, … range over Var, ‘P’ over Pred, ‘A’ over K-formulae, and ‘Γ’ over sets of K-formulae.

Let Kr be the class of all those quantificational formulas whose matrices are conjunctions of binary disjunctions of signed atomic formulas. Decision problems for subclasses of Kr do not invariably coincide with those for the corresponding classes of quantificational formulas with unrestricted matrices; for example, the ∀∃∀ prefix subclass of Kr is solvable, but the full ∀∃∀ class is not ([AaLe],- [KMW]). Moreover, the straightforward encodings of automata which have been used to show the unsolvability of various subclasses of Kr ([Aa], [Bö], [AaLe]) yield but little information about signature subclasses, i.e. subclasses determined by the number and degrees of the predicate letters occurring in a formula. By a new and highly complex construction Theorem 1 establishes the best possible result on classification by signature.

Theorem 1. Let C be the class of all formulas in Kr with a single predicate letter, which is dyadic; then C is a reduction class for satisfiability.

Thus a signature subclass of Kr is solvable just in case the corresponding class of unrestricted quantificational formulas is solvable, to wit, just in case no predicate letter of degree exceeding one may occur. To obtain a richer classification by signature we consider further restrictions on the matrix. Let Or be the class of formulas in Kr having disjunctive normal forms with only two disjuncts. Theorem 2 sharpens Orevkov's proof of the unsolvability of Or ([Or]; see also [LeGo]).

Theorem 2. Let D be the class of formulas in Or with just two predicate letters, both pentadic; then D is a reduction class for satisfiability.

This paper answers some questions which naturally arise from the Spector-Gandy proof of their theorem that the π1 1 sets of natural numbers are precisely those which are defined by a Σ1 1 formula over the hyperarithmetic sets. Their proof used hierarchies on recursive linear orderings (H-sets) which are not well orderings. (In this respect they anticipated the study of nonstandard models of set theory.) The proof hinged on the following fact. Let e be a recursive linear ordering. Then e is a well ordering if and only if there is an H-set on e which is hyperarithmetic. It was implicit in their proof that there are recursive linear orderings which are not well orderings, on which there are H-sets. Further information on such nonstandard H-sets (often called pseudohierarchies) can be found in Harrison [4]. It is natural to ask: on which recursive linear orderings are there H-sets?

In Friedman [1] it is shown that there exists a recursive linear ordering e that has no hyperarithmetic descending sequences such that no H-set can be placed on e. In [1] it is also shown that if e is a recursive linear ordering, every point of which has an immediate successor and either has finitely many predecessors or is finitely above a limit point (heretofore called adequate) such that an H-set can be placed on e, then e has no hyperarithmetic descending sequences. In a related paper, Friedman [2] shows that there is no infinite sequence xn of codes for ω-models of the arithmetic comprehension axiom scheme such that each x n+ 1 is a set in the ω-model coded by xn , and each x n+1 is the unique solution of P(xn , x n+1) for some fixed arithmetic P.

Craig showed in [3] that Beth's theorem fails for 2nd order logic. This is so since the syntax for 2nd order logic can be represented in the structure of natural numbers and itself can be characterized in 2nd order logic. Thus the satisfaction relation S is implicitly definable yet not explicitly definable (by a Tarski diagonal argument) and Beth fails. Kreisel pointed out in a review of Craig's paper that a similar argument shows that Beth's theorem fails for ω-logic or for any logic whose syntax can be represented in and in which can be characterized. In this paper we use the abstract model-theoretic notions of truth adequacy, truth-maximality and truth-completeness developed by Feferman in [4] to prove the following generalization of Craig's result: If L is a truth-complete logic, then no structure in which the syntax is represented is L-characterizable. This is applied via a corollary to give new examples of failures of Beth's theorem. Another way to construe this result is that truth-complete logics L have the desirable model-theoretic property (see [5]) that the structure in which the syntax is represented is not generalized finite (interpreting generalized finite as L-characterizability).

Since the basic notions of [4] are defined in terms of many-sorted logics, we will consider all our logics to be many-sorted.

The notions of sets of reals being κ-Souslin (κ a cardinal) and admitting a λ-scale (λ an ordinal) are due respectively to D. A. Martin and Y.N. Moschovakis. A set is ω-Souslin if and only if it is Σ1 1 (analytic). We show that a set is ω-Souslin if and only if it admits an (ω + l)-scale. Jointly with Martin and Solovay we show that if κ is uncountable and has cofinality ω, then being κ-Souslin is equivalent to admitting a κ-scale. Our results together with those of Kechris give a new simultaneous characterization of Σ1 1 and Δ1 1 (Borel) sets (a set is Σ1 1 if it admits an (ω + 1)-scale and Δ1 1 if it admits an ω-scale) and determine completely the relation between the κ-Souslin sets and the sets admitting λ-scales.

The generic structures introduced by Abraham Robinson are now well established in model theory. Anyone who works with these structures soon begins to feel that, in some sense, the infinite generic structures are large and the finite generic structures are small. Here ‘large’ and ‘small’ do not refer to the cardinality of the structures but are used in the way we describe saturated structures as large and atomic structures as small. In this paper I isolate what I consider to be the large and small existentially closed (e.c.) structures and attempt to determine their role in the class of all e.c. structures.

In the usual context of model theory we are concerned with elementary embeddings and all formulas. E.c. structures are concerned with all embeddings and ∃1-formulas. Thus we need to look at ∃1-analogues of saturated and atomic structures. These ∃1-saturated structures have been around for some time; they are just the existentially universal structures. The corresponding ∃1-atomic structures are not new here (they appear in [8]) but I believe that this paper will add much to the understanding of them.

The bulk of this paper is in §§2 and 4. §2 deals with ∃1-atomic structures, and §4 is concerned with the ∃1-analogues of several results in Vaught's paper [12].

A similar program has been carried out by Pouzet in [6], [7], [8]; however, there the significance of e.c. and e.c. structures is not realized and consequently some of the simplicity of the situation is lost.

The main purpose of this paper is to show how partial recursive functions and isols can be used to generalize the following three well-known theorems of combinatorial theory.

(I) For every finite projective plane Π there is a unique number n such that Π has exactly n 2 + n + 1 points and exactly n 2 + n + 1 lines.

(II) Every finite projective plane of order n can be coordinatized by a finite planar ternary ring of order n. Conversely, every finite planar ternary ring of order n coordinatizes a finite projective plane of order n.

(III) There exists a finite projective plane of order n if and only if there exist n − 1 mutually orthogonal Latin squares of order n.

Throughout this paper, α will denote an admissible ordinal. Let (α) denote the lattice of α-r.e. sets, i.e., the lattice whose elements are the α-r.e. sets, and whose ordering is given by set inclusion. Call a set A ∈ (α)α*-finite if it is α-finite and has ordertype < α* (the Σ1-projectum of α). The α*-finite sets form an ideal of (α), and factoring (α) by this ideal, we obtain the quotient lattice *(α).

We will fix a language ℒ suitable for lattice theory, and discuss decidability in terms of this language. Two approaches have succeeded in making some progress towards determining the decidability of the elementary theory of (α). Each approach was first used by Lachlan for α = ω. The first approach is to relate the decidability of the elementary theory of (α) to that of a suitable quotient lattice of (α) by a congruence relation definable in ℒ. This technique was used by Lachlan [4, §1] to obtain the equidecidability of the elementary theories of (ω) and *(ω), and was generalized by us [6, Corollary 1.2] to yield the equidecidability of the elementary theories of (α) and *(α) for all α. Lachlan [3] then adopted a different approach.

Let α be an admissible ordinal, and let (α) denote the lattice of α-r.e. sets, ordered by set inclusion. An α-r.e. set A is α*-finite if it is α-finite and has ordertype less than α* (the Σ1 projectum of α). An a-r.e. set S is simple if (the complement of S) is not α*-finite, but all the α-r.e. subsets of are α*-finite. Fixing a first-order language ℒ suitable for lattice theory (see [2, §1] for such a language), and noting that the α*-finite sets are exactly those elements of (α), all of whose α-r.e. subsets have complements in (α) (see [4, p. 356]), we see that the class of simple α-r.e. sets is definable in ℒ over (α). In [4, §6, (Q22)], we asked whether an admissible ordinal α exists for which all simple α-r.e. sets have the same 1-type. We were particularly interested in this question for α = ℵ1 L (L is Gödel's universe of constructible sets). We will show that for all α which are regular cardinals of L (ℵ1 L is, of course, such an α), there are simple α-r.e. sets with different 1-types.

The sentence exhibited which differentiates between simple α-r.e. sets is not the first one which comes to mind. Using α = ω for intuition, one would expect any of the sentences “S is a maximal α-r.e. set”, “S is an r-maximal α-r.e. set”, or “S is a hyperhypersimple α-r.e. set” to differentiate between simple α-r.e. sets. However, if α > ω is a regular cardinal of L, there are no maximal, r-maximal, or hyperhypersimple α-r.e. sets (see [4, Theorem 4.11], [5, Theorem 5.1] and [1,Theorem 5.21] respectively). But another theorem of (ω) points the way.

In [4] Kleene gave a definition of recursive functionals of finite type. Later Sacks [5] and Harrington [2] gave definitions of recursion in normal functionals of finite type. These definitions, that Sacks and Harrington assumed equivalent as far as normal objects are concerned, are nevertheless very different: Kleene's definition is given in terms of an inductive definition; Sacks uses simultaneously a hierarchy (the S σ F's) and induction on the ordinals and on the type; Harrington's universe does not use the induction on the type but uses a hierarchy as Shoenfield [6]. In this paper we prove in detail that, as was expected, the three definitions are equivalent.

In this paper we prove that if L is a set of modal propositional formulas then FR(L) (the class of all frames in which every formula of L holds) is elementary, Δ-elementary or not ΣΔ-elementary. For single modal formulas the second of these cases does not occur.

The model theoretic terminology and results used here are from [1]. (The underlying first order language contains only one, binary, predicate letter in addition to the identity symbol.) We presuppose familiarity with the usual notions and notations of propositional modal logic. A structure for our first order language is called a frame. (So a frame is an ordered couple 〈W, R〉 with domain W and R a binary predicate on W, i.e. a subset of W × W.) A valuation V on F is a function from the set of proposition letters to the power set of W. Using the well-known Kripke truth definition V can be extended to a function from the set of all modal propositional formulas to the power set of W. A modal propositional formula φ holds in a frame F (= 〈W, R〉) if, for all V on F, V(φ) = W. Notation: FR(φ) for the class of all frames in which φ holds. For a set L of modal propositional formulas we define FR(L) as ⋂φ∈L FR(φ). Obviously both FR(L) and cFR(L) (the complement of FR(L)) are closed under isomorphisms.

An r-normal function is a strictly increasing continuous function from r to r where r is a regular ordinal > ω (identify an ordinal with the set of smaller ordinals). Given an r-normal function f one can form a sequence {f(x, −)} x<r of r-normal functions—the Veblen hierarchy [33] on f—as follows: f(0, −) = f and, for x > 0, f(x, −) enumerates in order {z ∣ f(y, z) = z for all y < x}, the common fixed points of the f(y, −)'s for y < x. In this paper we give as readable an exposition as we can of Veblen hierarchies and of Bachmann's and Isles's techniques in [3] and [15] of using higher finite number classes for forming sequences {f(x, −)} x<y where y > r of r-normal functions which extend the Veblen hierarchy on f. We will show how these sequences—Bachmann hierarchies—yield extremely natural constructive notations for ordinals in various initial segments of the second number class. We will also consider various other techniques for obtaining constructive ordinal notations and relate them to the notations obtained by Bachmann's and Isles's techniques. In particular, we will use these notations to characterize as directly and as usefully as we can various of Takeuti's systems of constructive ordinal notations, which he calls ordinal diagrams ([31], [32]).

The standard classes of a first-order theory T are certain classes of prenex T-sentences defined by restrictions on prefix, number of monadic, dyadic, etc. predicate variables, and number of monadic, dyadic, etc. operation variables. In [3] it is shown that, for any theory T, (1) the decision problem for any class of prenex T-sentences specified by such restrictions reduces to that for the standard classes, and (2) there are finitely many standard classes K 1, …, Kn such that any undecidable standard class contains one of K 1, …, Kn . These results give direction to the study of the decision problem.

Below T is predicate logic with identity and operation variables. The Main Theorem solves the decision problem for the standard classes admitting at least one operation variable.

The set theory AFC was introduced by Perlis in [2] and he noted that it both includes and is stronger than Ackermann's set theory. We shall give a relative consistency result for AFC.

AFC is obtained from Ackermann's set theory (see [2]) by replacing Ackermann's set existence schema with the schema

(where ϕ, ψ, are ∈-formulae, x is not in ψ, w is not in ϕ, y is y 1, …, yn, z is z 1, …, zm and all free variables are shown) and adding the axiom of choice for sets. Following [1], we say that λ is invisible in Rκ if λ < κ and we have

holding for every ∈-formula θ which has exactly two free variables and does not involve u or υ. The existence of a Ramsey cardinal implies the existence of cardinals λ and κ with λ invisible in Rκ, and Theorem 1.13 of [1] gives some further indications about the relative strength of the notion of invisibility.

Theorem. If there are cardinals λ and κ with λ invisible in Rκ, then AFC is consistent.

Proof. Suppose that λ is invisible in Rκ and we will show that 〈Rκ, Rλ, ∈〉 ⊧ AFC (Rλ being the interpretation of V, of course).

In [4] Curry raised the possibility that his system proposed in ξ15C of [3] might be inconsistent. In this paper this inconsistency is proved using a method also employed in [1].

From Curry's axiom ⊦LH, it follows that

holds for arbitrary X.

The other results from that are required are

Modus Ponens, and the Deduction Theorem for implication:

Assuming ⊦HA, we define as in [1]:

and let

where Y is the paradoxical (or fixed point) combinator.