A finite set of tiles (unit squares with colored edges) is said to tile the plane if there exists an arrangement of translated (but not rotated or reflected) copies of the squares which fill the plane in such a way that abutting edges of the squares have the same color. The problem of whether there exists a finite set of tiles which can be used to tile the plane but not in any periodic fashion was proposed by Hao Wang [9] and solved by Robert Berger [1]. Raphael Robinson [7] gives a more detailed history and a very economical solution to this and related problems; we will assume that the reader is familiar with §4 of [7]. In 1971, Dale Myers asked whether there exists a finite set of tiles which can tile the plane but not in any recursive fashion. If we make an additional restriction (called the origin constraint) that a given tile must be used at least once, then the positive answer is given by the main theorem of this paper. Using the Turing machine constructed here and a more complicated version of Berger and Robinson's construction, Myers [5] has recently solved the problem without the origin constraint.

Given a finite set of tiles T 1, …, Tn , we can describe a tiling of the plane by a function f of two variables ranging over the integers. f(i, j) = k specifies that the tile Tk is to be placed at the position in the plane with coordinates (i, j). The tiling will be said to be recursive if f is a recursive function.

We prove, in ZFC, the existence of a definable, countably saturated elementary extension of the reals.

The reader of Part VI will have noticed that among the set-theoretic models considered there some models were missing which were announced in Part II for certain proofs of independence. These models will be supplied now.

Mainly two models have to be constructed: one with the property that there exists a set which is its own only element, and another in which the axioms I–III and VII, but not Va, are satisfied. In either case we need not satisfy the axiom of infinity. Thereby it becomes possible to set up the models on the basis of only I–III, and either VII or Va, a basis from which number theory can be obtained as we saw in Part II.

On both these bases the Π0-system of Part VI, which satisfies the axioms I–V and VII, but not VI, can be constructed, as we stated there. An isomorphic model can also be obtained on that basis, by first setting up number theory as in Part II, and then proceeding as Ackermann did.

Let us recall the main points of this procedure.

For the sake of clarity in the discussion of this and the subsequent models, it will be necessary to distinguish precisely between the concepts which are relative to the basic set-theoretic system, and those which are relative to the model to be defined.

Treating the existential quantification ∃νi as a diamond ♢i and the identity νi = νj as a constant δij, we study restricted versions of first order logic as if they were modal formalisms. This approach is closely related to algebraic logic, as the Kripke frames of our system have the type of the atom structures of cylindric algebras; the full cylindric set algebras are the complex algebras of the intended multidimensional frames called cubes.

The main contribution of the paper is a characterization of these cube frames for the finite-dimensional case and, as a consequence of the special form of this characterization, a completeness theorem for this class. These results lead to finite, though unorthodox, derivation systems for several related formalisms, e.g. for the valid n-variable first order formulas, for type-free valid formulas and for the equational theory of representable cylindric algebras. The result for type-free valid formulas indicates a positive solution to Problem 4.16 of Henkin, Monk and Tarski [16].

We define a propositionally quantified intuitionistic logic Hπ+ by a natural extension of Kripke's semantics for propositional intuitionistic logic. We then show that Hπ+ is recursively isomorphic to full second order classical logic. Hπ+ is the intuitionistic analogue of the modal systems S5π+, S4π+, S4.2π+, K4π+, Tπ+, Kπ+ and Bπ+, studied by Fine.

We show that several theorems on ordinal bounds in different parts of logic are simple consequences of a basic result in the theory of global inductive definitions.

The logic of quantification is conveniently developed in terms of the so-called quantificational schemata. The atomic quantificational schemata consist each of a schematic predicate letter (‘F’, ‘G’, etc.) attached to one or more quantifiable variables (‘x’, ‘y’, etc.). Compound quantificational schemata are built up of these atomic ones by means of truth functions and quantifiers. The business of the logic of quantification then becomes that of spotting the valid schemata: the ones that are true under all interpretations.

The logic of quantification is well understood and well behaved. Proof procedures in this domain are known, thanks to Gödel, to be complete. If decision procedures are not known for the whole of this part of logic, still we do know, thanks to Church, that they cannot exist. And we know how to streamline our proof procedure for quick success in the average case where proof is possible. Because of all this, there is a premium on adapting special theories, of special subject matters, to this general mold. This means representing the truths of the special theory as a class of quantificational schemata with interpreted predicate letters and a chosen universe of discourse. Once a theory has been thus regimented, it is a standard theory (in approximately Tarski's sense of the term).

This paper presents three generalizations of the Second Incompleteness Theorem of Gödel (see [2]) which apply to a broader class of formal systems than previous generalizations (as, e.g., the generalization in [1]).

The content of all three of our results is that it is versions of the third derivability condition of Hilbert and Bernays (see p. 286 of [3]) which are crucial to Gödel's theorem. The second derivability condition plays some role in certain logics, but even there, only in the far weaker form of a definability condition (see Theorem 2).

The elimination of the first derivability condition allows the application of the Consistency Theorem to cut-free logics which cannot prove that they are closed under cut.

It is Theorem 1 which will probably have primary interest for readers who are not concerned with technical proof theory or with foundations, for it treats logics with quantifiers, and in that case one can dispose entirely of the first and second derivability conditions. The derivation of this result requires a “new twist” on old arguments. Specifically, we use a new variation on the standard self-referential lemma (this is Lemma 5.1 of [l]) to obtain a somewhat different self-referential construction than has previously been employed (this is our lemma in §2 below). With this new construction, we consider the sentence φ which “expresses” the fact: “My negation is provable.” Then, using hypotheses associated with Gödel's First Incompleteness Theorem, one shows that ⊢¬φ is impossible in a consistent logic. Finally, using only a version of the third Hilbert-Bernays derivability condition, one shows that the provability of the consistency statement implies ⊢¬φ, and hence that consistency is unprovable.

As we know, the system of material implication is led into an inconsistency by the Russellian class, defined as λx. Nϵxx. This class, however, does no harm to many other systems, for example, the three-valued system L3 given by J. Łukasiewicz. A natural question is whether or not there exist classes which affect some of these other logical systems. The main result of the present paper is to answer this question affirmatively. At the end of this paper we point out that the interpretation given by J. Łukasiewicz for the system L3 is not satisfactory, and propose a new interpretation.

B. Russell deduced the mentioned inconsistency by the aid of the notion of negation. Later on, H. B. Curry pointed out that we could get the same result without the aid of that notion. None of these results affects the system L3 and other similar systems. But these systems may be involved.

To show this, we need the following definitions.

A function of two variables Cpq will be called an implication when the following “implication rule” is valid:

Under this definition we should note that material equivalence Epq, for example, is an implication.

Let C be such an implication. Then the symbol “(Cp)iq” is defined recursively by

The class an is defined as λx. (Cϵxx)np, where p is a propositional variable. The rule of absorption of order n, denoted by (An), is:

Let T be a countable complete theory and C(T) the category whose objects are the models of T and morphisms are the elementary maps. The main object of this paper will be the study of C(T). The idea that a better understanding of the category may give us model theoretic information about T is quite natural: The (semi) group of automorphisms (endomorphisms) of a given structure is often a powerful tool for studying this structure. But certainly, one of the very first questions to be answered is: “to what extent does this category C(T) determine T?”

There is some obvious limitation: for example let T0 be the theory of infinite sets (in a language containing only =) and T1 the theory, in the language ( =, U(ν0),f(ν0)) stating that:

(1) U is infinite.

(2)f is a bijective map from U onto its complement.

It is quite easy to see that C(T0) is equivalent to C(T1). But, in this case, T0 andT1 can be “interpreted” each in the other. To make this notion of interpretation precise, we shall associate with each theory T a category, loosely denoted by T, defined as follows:

(1) The objects are the formulas in the given language.

(2) The morphisms from into are the formulas such that

(i.e. f defines a map from ϕ into ϕ; two morphisms defining the same map in all models of T should be identified).

We show in ZFC that if there is no proper class inner model with a Woodin cardinal, then there is an absolutely definable core model that is close to V in various ways.

The purpose of this paper is to prove two theorems and a conjecture (Conjecture II) announced in section 15 an earlier paper of the author's (cited as “CT”), and to compare them briefly with related results of Specker. Familiarity with both papers is assumed; the terminology of the former is used throughout. On two points however clarification of the usage of CT is in order, and to this chore we must first proceed.

A half-section is the lower half of a Dedekind cut; if the cut is rational, the half section is to include the rational corresponding to the real defined by the cut. A whole-section is the relation which holds between any member of the lower and any member of the upper half of some Dedekind cut. If the cut is rational the corresponding rational is to be a member of both halves.

A real number α is said to be approximate in K to any required number of decimal places if it is possible to define the predicates ‘x < α’, ‘x ≤ α’, ‘x > α’, and ‘x ≥ α’ (x rational) in K. In view of section 7 of CT this will mean that every true inequation between α and a terminating decimal will be provable in K.

The extreme interest of set theorists in the notion of “closed unbounded set” is epitomized in the following well-known theorem:

Theorem A. For any regular cardinal κ > ω, the intersection of any two closed unbounded subsets of κ is closed and unbounded.

The proof of this theorem is easy and in fact yields a stronger result, namely that for any uncountable regular cardinal κ the intersection of fewer than κ many closed unbounded sets is closed and unbounded. Thus, if, for κ a regular uncountable cardinal, we let denote {A ⊆ κ ∣ A contains a closed unbounded subset}, then, for any such κ, is a κ-additive nonprincipal filter on κ.

Now what about the possibility of being an ultrafilterκ It is routine to see that this is impossible for κ > ℵ1. However, for κ = ℵ1 the situation is different. If were an ultrafilter, ℵ1 would be a measurable cardinal. As is well-known this is impossible if we assume the axiom of choice; however if ZF + “there exists a measurable cardinal” is consistent, then so is ZF + “ℵ1 is a measurable cardinal” [2]. Furthermore, under the assumption of certain set theoretic axioms (such as the axiom of determinateness or various infinite exponent partition relations) can be proven to be an ultrafilter. (See [3] and [5].)

In his well-known paper [11], Post founded a general theory of recursively enumerable sets, which had its metamathematical source in questions about the decision problem for deducibility in formal systems. However, in centering attention on the notion of degree of unsolvability, Post set a course for his theory which has rarely returned to this source. Among exceptions to this tendency we may mention, as being closest to the problems considered here, the work of Kleene in [8] pp. 298–316, of Myhill in [10], and of Uspenskij in [15]. It is the purpose of this paper to make some further contributions towards bridging this gap.

From a certain point of view, it may be argued that there is no real separation between metamathematics and the theory of recursively enumerable sets. For, if the notion of formal system is construed in a sufficiently wide sense, by taking as ‘axioms’ certain effectively found members of a set of ‘formal objects’ and as ‘proofs’ certain effectively found sequences of these objects, then the set of ‘provable statements’ of such a system may be identified, via Gödel's numbering technique, with a recursively enumerable set; and conversely, each recursively enumerable set is identified in this manner with some formal system (cf. [8] pp. 299–300 and 306). However, the pertinence of Post's theory is no longer clear when we turn to systems formalized within the more conventional framework of the first-order predicate calculus. It is just this restriction which serves to clarify the difference in spirit of the two disciplines.

We shall apply some of the results of Jensen [4] to deduce new combinatorial consequences of the axiom of constructibility, V = L. We shall show, among other things, that if V = L then for each cardinal λ there is a set A ⊆ λ such that neither A nor λ – A contain a closed set of type ω1. This is an extension of a result of Silver who proved it for λ = ω2, providing a partial answer to Problem 68 of Friedman [2].

The main results of this paper were obtained independently by both authors.

If λ is an ordinal, E is said to be Mahlo (or stationary) in λ, if λ – E does not contain a closed cofinal subset of λ.

Consider the statements:

(J1) There is a class E of limit ordinals and a sequence Cλ defined on singular limit ordinals λ such that

(i) E ⋂ μ is Mahlo in μ for all regular > ω;

(ii) Cλ is closed and unbounded in λ;

(iii) if γ < is a limit point of Cλ, then γ is singular, γ ∉ E and Cγ = γ ⋂ Cλ.

For each infinite cardinal κ:

(J2,κ) There is a set E ⊂ κ+ and a sequence Cλ(Lim(λ), λ < κ+) such that

(i) E is Mahlo in κ+;

(ii) Cλ is closed and unbounded in λ;

(iii) if cf(λ) < κ, then card Cλ < κ;

(iv) if γ < λ is a limit point of Cλ then γ ∉ E and Cγ = ϣ ⋂ Cλ.

In this paper I give a characterization of the closed fragment of the provability logic of I Δ0 + EXP with a propositional constant for EXP. In three appendices many details on arithmetization are provided.

By means of models in toposes of C-sets (where C is a small category), necessary conditions are found for the minimum quantified extension of a propositional (intermediate, modal) logic to be complete with respect to Kripke semantics; in particular, many well-known systems turn out to be incomplete.

In this paper we introduce a new notion of realizability for intuitionistic arithmetic in all finite types. The notion seems to us to capture some of the intuition underlying both the recursive realizability of Kjeene [5] and the semantics of Kripke [7]. After some preliminaries of a syntactic and recursion-theoretic character in §1, we motivate and define our notion of realizability in §2. In §3 we prove a soundness theorem, and in §4 we apply that theorem to obtain new information about provability in some extensions of intuitionistic arithmetic in all finite types. In §5 we consider a special case of our general notion and prove a kind of reflection theorem for it. Finally, in §6, we consider a formalized version of our realizability notion and use it to give a new proof of the conservative extension theorem discussed in Goodman and Myhill [4] and proved in our [3]. (Apparently, a form of this result is also proved in Mine [13]. We have not seen this paper, but are relying on [12].) As a corollary, we obtain the following somewhat strengthened result: Let Σ be any extension of first-order intuitionistic arithmetic (HA) formalized in the language of HA. Let Σω be the theory obtained from Σ by adding functionals of finite type with intuitionistic logic, intensional identity, and axioms of choice and dependent choice at all types. Then Σω is a conservative extension of Σ. An interesting example of this theorem is obtained by taking Σ to be classical first-order arithmetic.

The main theorem of this paper states that if R is a ring and is a totally transcendental R-module, then has a unique decomposition as a direct sum of indecomposable R-modules. Natural examples of totally transcendental modules are injective modules over noetherian rings, artinian modules over commutative rings, projective modules over left-perfect, right-coherent rings, and arbitrary modules over Σ – α-gens rings. Therefore, our decomposition theorem yields as special cases the purely algebraic unique decomposition theorems for these four classes of modules due to Matlis; Warfield; Mueller, Eklof, and Sabbagh; and Shelah and Fisher. These results and a number of other corollaries about totally transcendental modules are covered in §1. In §2, I show how the results of § 1 can be used to give an improvement of Baur's classification of ω-categorical modules over countable rings. In §3, the decomposition theorem is used to study modules with quantifier elimination over noetherian rings.

The goals of this section are to prove the decomposition theorem and to derive some of its immediate corollaries. I will begin with some notational conventions. R will denote a ring with an identity element. LR is the language of left R-modules described in [4, p. 251] and TR is the theory of left R-modules. “R-module” will mean “unital left R-module”. A formula will mean an LR-formula.