Let P0, …,Pq be predicates, of which at least one has two or more places. By a formula in P0, …, Pq (or simply a formula, when the list P0, …, Pq is fixed, as in this section) is meant any formula whose only symbols, other than sentential connectives, quantifiers, and (individual) variables, are among P0, …, Pq. A realization (or possible model) of such a formula is a system where A is a non-empty set and each Pk is a relation among the elements of A, having the same number of places as Pk.

A boolean algebra is shown to be completely representable if and only if it is atomic, whereas it is shown that neither the class of completely representable relation algebras nor the class of completely representable cylindric algebras of any fixed dimension (at least 3) are elementary.

In this paper, following an idea of Christophe Chalons, I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence φ holding in some forcing extension Vℙ and all subsequent extensions Vℙ*ℚ holds already in V. It follows, in fact, that such sentences must also hold in all forcing extensions of V. In modal terms, therefore, the Maximality Principle is expressed by the scheme (◊ □ φ) ⇒ □ φ, and is equivalent to the modal theory S5. In this article, I prove that the Maximality Principle is relatively consistent with ZFC. A boldface version of the Maximality Principle, obtained by allowing real parameters to appear in φ, is equiconsistent with the scheme asserting that Vδ ≺ V for an inaccessible cardinal δ, which in turn is equiconsistent with the scheme asserting that ORD is Mahlo. The strongest principle along these lines is □ , which asserts that holds in V and all forcing extensions. From this, it follows that 0# exists, that x# exists for every set x, that projective truth is invariant by forcing, that Woodin cardinals are consistent and much more. Many open questions remain.

The infinite-valued statement calculus to which this paper refers is that of Łukasiewicz [10], whose axiomatization was proved complete in [5]. In [9], Rutledge extended this system to include predicates and quantifiers2 and presented a deductively complete set of axioms for the monadic predicate calculus. This paper represents an attempt to axiomatize the full predicate calculus; for the proposed axiomatization, a property akin to but weaker than completeness is proved. An attempt to prove full completeness along similar lines failed; it has since been shown [11] that the set of valid formulas of the infinite-valued predicate calculus is not recursively enumerable. The method of this paper was suggested by Professor J. Barkley Rosser.

One of the standard ways of proving the consistency of additional hypotheses with the basic axioms of an axiom system is by the construction of what may be described as ‘inner models.’ By starting with a domain of individuals assumed to satisfy the basic axioms an inner model is constructed whose domain of individuals is a certain subset of the original individual domain. If such an inner model can be constructed which satisfies not only the basic axioms but also the particular additional hypothesis under consideration, then this affords a proof that if the basic axiom system is consistent then so is the system obtained by adding to this system the new hypothesis. This method has been applied to axiom systems for set theory by many authors, including v. Neumann (4), Mostowski (5), and more recently Gödel (1), who has shown by this method that if the basic axioms of a certain axiomatic system of set theory are consistent then so is the system obtained by adding to these axioms a strong form of the axiom of choice and the generalised continuum hypothesis. Having been shown in this striking way the power of this method it is natural to inquire whether it has any limitations or whether by the construction of a sufficiently ingenious inner model one might hope to decide other outstanding consistency questions, such as the consistency of the negations of the axiom of choice and continuum hypothesis. In this and two following papers we prove some general theorems concerning inner models for a certain axiomatic system of set theory which lead to the result that as far as a fairly large family of inner models are concerned this method of proving consistency has been exhausted, that no essentially new consistency results can be obtained by the use of this kind of model.

The complexity of a computable function can be measured by considering the time or space required to compute its values. Particular notions of time and space arising from variants of Turing machines have been investigated by R. W. Ritchie [14], Hartmanis and Stearns [8], and Arbib and Blum [1], among others. General properties of such complexity measures have been characterized axiomatically by Rabin [12], Blum [2], Young [16], [17], and McCreight and Meyer [10].

In this paper the speed-up and super-speed-up theorems of Blum [2] are generalized to speed-up by arbitrary total effective operators. The significance of such theorems is that one cannot equate the complexity of a computable function with the running time of its fastest program, for the simple reason that there are computable functions which in a very strong sense have no fastest programs.

Let φi be the ith partial recursive function of one variable in a standard Gödel numbering of partial recursive functions. A family Φ0, Φ1, … of functions of one variable is called a Blum measure on computation providing

(1) domain (φi) = domain (Φi), and

(2) the predicate [Φi(x) = m] is recursive in i, x and m.

Typical interpretations of Φi(x) are the number of steps required by the ith Turing machine (in a standard enumeration of Turing machines) to converge on input x, the space or number of tape squares required by the ith Turing machine to converge on input x (with the convention that Φi(x) is undefined even if the machine fails to halt in a finite loop), and the length of the shortest derivation of the value of φi(x) from the ith set of recursive equations.

We prove that by showing that for any set C not of PA-degree and any set A, there exists an infinite subset G of A or such that G ⊕ C is also not of PA-degree.

By a logical measure (for a given language) we mean a syntactically defined function which associates some value with each well-formed formula of the language. Various such logical measures have played a fundamental role in the development of Logic and the Philosophy of Science. The purpose of this paper is to define a logical measure which has much wider applications than measures so far studied.

The new measure has two fundamental advantages, which will be referred to throughout the paper. First of all it can be applied to more (and richer) languages than the older measures. Most of the measures now in use are restricted to the first-order functional calculus, and frequently even to a first-order calculus with one-place predicates only. The measure to be defined will be applicable to richer languages as well, e.g., to functional calculi of all finite orders. But even as far as the first-order functional calculus is concerned, the new measure has a great advantage: We do not have to require that the atomic sentences of the calculus be independent. The previous measures depended in their construction on the requirement that “the basic statements of the language express independent facts.”

By an atomic sentence we mean a well-formed formula formed by applying an h-place primitive predicate to h individuals (a well-formed formula no part of which is well-formed); by a permissible conjunction we mean a conjunction of atomic sentences and negations of other atomic sentences; and the requirement of independence is that an atomic sentence (or its negation) is logically implied by a permissible conjunction only if it (its negation) is one of the components of the conjunction.

Beth's Definability Theorem, and consequently the Interpolation Lemma, fail for the version of quantified S5 that is presented in Kripke's [6]. These failures persist when the constant domain axiom-scheme ∀x□φ ≡ □∀xφ is added to S5 or, indeed, to any weaker extension of quantificational K.

§1 reviews some standard material on quantificational modal logic. This is in contrast to quantified intermediate logics for, as Gabbay [6] has shown, the Interpolation Lemma holds for the logic CD with constant domains and for several of its extensions. §§2—4 establish the negative results for the systems based upon S5. §5 establishes a more general negative result and, finally, §6 considers some positive results and open problems. A basic knowledge of classical and modal quantificational logic is presupposed.

Let me briefly review the relevant model theory for quantified modal logic. Further details can be found in [3] or [7].

The language is obtained from the language for classical first-order logic with identity by adding a unary operator □ for necessity. The atomic formula ‘Ex’ is used as an abbreviation for ‘∃y(y = x)’ and may be read as ‘x exists’.

The purpose of the present paper is to give a new, simple proof of the theorem of M. Davis, H. Putnam and J. Robinson [1961], which states that every recursively enumerable relation A(a1, …, an) is exponential diophantine, i.e. can be represented in the form

where a1 …, an, x1, …, xm range over natural numbers and R and S are functions built up from these variables and natural number constants by the operations of addition, A + B, multiplication, AB, and exponentiation, AB. We refer to the variables a1,…,an as parameters and the variables x1 …, xm as unknowns.

Historically, the Davis, Putnam and Robinson theorem was one of the important steps in the eventual solution of Hilbert's tenth problem by the second author [1970], who proved that the exponential relation, a = bc, is diophantine, and hence that the right side of (1) can be replaced by a polynomial equation. But this part will not be reproved here. Readers wishing to read about the proof of that are directed to the papers of Y. Matijasevič [1971a], M. Davis [1973], Y. Matijasevič and J. Robinson [1975] or C. Smoryński [1972]. We concern ourselves here for the most part only with exponential diophantine equations until §5 where we mention a few consequences for the class NP of sets computable in nondeterministic polynomial time.

PA is Peano arithmetic. The formula InterpPA(α, β) is a formalization of the assertion that the theory PA + α interprets the theory PA + β (the variables α and β are intended to range over codes of sentences of PA). We extend Solovay's modal analysis of the formalized provability predicate of PA, PrPA(x), to the case of the formalized interpretability relation InterpPA(x, y). The relevant modal logic, in addition to the usual provability operator ‘□’, has a binary operator ‘⊳’ to be interpreted as the formalized interpretability relation. We give an axiomatization and a decision procedure for the class of those modal formulas that express valid interpretability principles (for every assignment of the atomic modal formulas to sentences of PA). Our results continue to hold if we replace the base theory PA with Zermelo-Fraenkel set theory, but not with Gödel-Bernays set theory. This sensitivity to the base theory shows that the language is quite expressive. Our proof uses in an essential way earlier work done by A. Visser, D. de Jongh, and F. Veltman on this problem.

The main result of this paper is Theorem 3.1 which is a criterion for weak o-minimality of a linearly ordered structure in terms of realizations of 1-types. Here we also prove some other properties of weakly o-minimal structures. In particular, we characterize all weakly o-minimal linear orderings in the signature {<, =} . Moreover, we present a criterion for density of isolated types of a weakly o-minimal theory. Lastly, at the end of the paper we present some remarks on the Exchange Principle for algebraic closure in a weakly o-minimal structure.

In Anderson and Belnap [8] there was developed a semantics for first degree entailments (fde), i.e., entailments A → B between formulas A and B involving only truth-functions (defined in terms of “or” and “not”) and quantifiers. The key ideas were (i) the notion of a frame ⟨P, FIP, I⟩, where P is a set of (intensional) propositions closed under negation and multiple disjunction, I is a domain of individuals, and FIP is the set of functions from I into P; and (ii) the semantic relation of cons (consequence), as obtaining between a set of propositions taken conjunctively, and a set taken disjunctively; and (iii) the notion of an atomic frame, i.e., a frame generated by a set of propositions X closed under negation, such that for any disjoint subclasses Y and Z of X, Y does not bear cons to Z.

The quantified relevant logic RQ is given a new semantics in which a formula ∀xA is true when there is some true proposition that implies all x-instantiations of A. Formulae are modelled as functions from variable-assignments to propositions, where a proposition is a set of worlds in a relevant model structure. A completeness proof is given for a basic quantificational system QR from which RQ is obtained by adding the axiom EC of ‘extensional confinement’: ∀x(A ⋁ B) → (A ⋁ ∀xB), with x not free in A. Validity of EC requires an additional model condition involving the boolean difference of propositions. A QR-model falsifying EC is constructed by forming the disjoint union of two natural arithmetical structures in which negation is interpreted by the minus operation.

A standard set of axioms for the intuitionist propositional calculus is the following (we associate to the right, and use the convention that ⊃ binds less strongly than the other connectives).

Below are collected some simple results on the theory of free choice sequences [4] or infinitely proceeding sequences (ips) as they are called in [5]. These results are not sufficient to settle the completeness problems for Heyting's predicate calculus [4], as formulated in [12], neither in the strong nor in the weak sense. They are published because they lead to intuitionistically valid versions of completeness proofs which have appeared in the literature, particularly [16], [17], [18], and, with certain reservations, [1].

The problems considered below (except in §8) differ from those of [12] in the following respect: In [12] we were mainly concerned with formulae of Heyting's predicate calculus which were not even classically provable, and showed that the calculus was complete with respect to certain classes of these formulae. The novel feature was that we established this completeness by means of intuitionistically valid methods, in fact methods which can be formalized in Heyting's arithmetic. Here we are primarily concerned with formulae which are classically, but not intuitionistically, provable. As pointed out in [12], the nature of the completeness problem for such formulae is totally different: the class of predicates which provides the required counterexamples must come from a system (with an intuitionistically acceptable interpretation) which, unlike Heyting's arithmetic, is not a subsystem of the corresponding classical system. An example of such a system is an extension FC, given below, of Heyting's formalization of the theory of free choices.

Dummett's LC [1] is a system which characterizes all formulas of the propositional calculus which are valid in every chain (for definitions and notation see the first section of [2]). An L-algebra is a Heyting algebra in which (x → y) + (y → x) = 1 for all x, y. L-algebras bear the same relation to LC as Boolean algebras to the classical propositional calculus and Heyting algebras to the intuitionist propositional calculus.

A. H. Lachlan [2] and C. E. M. Yates [4] independently showed that minimal pairs of recursively enumerable (r.e.) degrees exist. Lachlan and Richard Ladner have shown (unpublished) that there is no uniform method for producing a minimal pair of r.e. degrees below a given nonzero r.e. degree. It is not known whether every nonzero r.e. degree bounds a r.e. minimal pair, but in the present paper it is shown (uniformly) that every high r.e. degree bounds a r.e. minimal pair. (A r.e. degree is said to be high if it contains a high set in the sense of Robert W. Robinson [3].)

Theorem. Let a be a recursively enumerable degree for which a′ = 0″. Then there are recursively enumerable degrees b0 and b1 such that0 < bi < a for each i ≤ 1, and b0 ⋂ b1 = 0.

The proof is based on the Lachlan minimal r.e. pair construction. For notation see Lachlan [2] or S. B. Cooper [1].

By Robinson [3] we can choose a r.e. representative A of the degree a, with uniformly recursive tower {As, ∣ s ≥ 0} of finite approximations to A, such that CA dominates every recursive function where

We define, stage by stage, finite sets Bi,s, i ≤ 1, s ≥ 0, in such a way that Bi, s + 1 ⊇ Bi,s for each i, s, and {Bi,s ∣ i ≤ 1, s ≥ 0} is uniformly recursive.

This paper is motivated by the question whether there exists a logic capturing polynomial time computation over unordered structures. We consider several algorithmic problems near the border of the known, logically defined complexity classes contained in polynomial time. We show that fixpoint logic plus counting is stronger than might be expected, in that it can express the existence of a complete matching in a bipartite graph. We revisit the known examples that separate polynomial time from fixpoint plus counting. We show that the examples in a paper of Cai, Fürer, and Immerman, when suitably padded, are in choiceless polynomial time yet not in fixpoint plus counting. Without padding, they remain in polynomial time but appear not to be in choiceless polynomial time plus counting. Similar results hold for the multipede examples of Gurevich and Shelah, except that their final version of multipedes is, in a sense, already suitably padded. Finally, we describe another possible candidate, involving determinants, for the task of separating polynomial time from choiceless polynomial time plus counting.

Frege's development of the theory of arithmetic in his Grundgesetze der Arithmetik has long been ignored, since the formal theory of the Grundgesetze is inconsistent. His derivations of the axioms of arithmetic from what is known as Hume's Principle do not, however, depend upon that axiom of the system—Axiom V—which is responsible for the inconsistency. On the contrary, Frege's proofs constitute a derivation of axioms for arithmetic from Hume's Principle, in (axiomatic) second-order logic. Moreover, though Frege does prove each of the now standard Dedekind-Peano axioms, his proofs are devoted primarily to the derivation of his own axioms for arithmetic, which are somewhat different (though of course equivalent). These axioms, which may be yet more intuitive than the Dedekind-Peano axioms, may be taken to be “The Basic Laws of Cardinal Number”, as Frege understood them.

Though the axioms of arithmetic have been known to be derivable from Hume's Principle for about ten years now, it has not been widely recognized that Frege himself showed them so to be; nor has it been known that Frege made use of any axiomatization for arithmetic whatsoever. Grundgesetze is thus a work of much greater significance than has often been thought. First, Frege's use of the inconsistent Axiom V may invalidate certain of his claims regarding the philosophical significance of his work (viz., the establish may invalidate certain of his claims regarding the philosophical significance of his work (viz., the establishment of Logicism), but it should not be allowed to obscure his mathematical accomplishments and his contribution to our understanding of arithmetic. Second, Frege's knowledge that arithmetic is derivable from Hume's Principle raises important sorts of questions about his philosophy of arithmetic. For example, “Why did Frege not simply abandon Axiom V and take Hume's Principle as an axiom?”