Eine auf neue Art und mit neuen symbolischen Mitteln durchgeführte Analyse des Prozesses der Prädikation ergibt, daβ das für die logisch-mengentheoretischen Paradoxa nach Art des Russellschen essentielle Moment weder das der Selbstprädikation als solches ist noch irgendetwas, was man anderweitig bisher für sie verantwortlich gemacht hat, sondern vielmehr der Umstand, daβ der Algorithmus der Prädikation ebensowohl wie zu einem natürlichen Abschluβ auch auf einen unendlichen Regreβ führen kann. Damit werden die genannten Paradoxa als den bekannten erledigten der mathematischen unendlichen Prozesse wesentlich gleichartig und der gleichen Behandlung zugänglich erkannt. Das neue Kalkülmittel der limitierten Variablen bewirkt – ohne irgendwelche metalogische Regulative ad hoc – die automatische Sicherung des logischen Kalküls gegen die früheren Verführungen.

A mapping u: X → Y between metric spaces is sequentially continuous if for each sequence (xn) converging to x ∈ X, (u(xn)) converges to u(x). It is well known in classical mathematics that a sequentially continuous mapping between metric spaces is continuous; but, as all proofs of this result involve the law of excluded middle, there appears to be a constructive distinction between sequential continuity and continuity. Although this distinction is worth exploring in its own right, there is another reason why sequential continuity is interesting to the constructive mathematician: Ishihara [8] has a version of Banach's inverse mapping theorem in functional analysis that involves the sequential continuity, rather than continuity, of the linear mappings; if this result could be upgraded by deleting the word “sequential”, then we could prove constructively the standard versions of the inverse mapping theorem and the closed graph theorem.

Troelstra [9] showed that in Brouwer's intuitionistic mathematics (INT) a sequentially continuous mapping on a separable metric space is continuous. On the other hand, Ishihara [6, 7] proved constructively that the continuity of sequentially continuous mappings on a separable metric space is equivalent to a certain boundedness principle for subsets of ℕ; in the same paper, he showed that the latter principle holds within the recursive constructive mathematics (RUSS) of the Markov School. Since it is not known whether that principle holds within Bishop's constructive mathematics (BISH), of which INT and RUSS are models and which can be regarded as the constructive core of mathematics, the exploration of sequential continuity within BISH holds some interest.

Let be a well-founded model of ZFC whose class of ordinals has uncountable cofinality, such that has a Σn end extension for each n ∈ ω. It is shown in Theorem 1.1 that there is such a model which has no elementary end extension. In the process some interesting facts about topless end extensions (those with no least new ordinal) are uncovered, for example Theorem 2.1: If is a well-founded model of ZFC, such that has uncountable cofinality and has a topless Σ3 end extension, then has a topless elementary end extension and also a well-founded elementary end extension, and contains ordinals which are (in ) highly hyperinaccessible. In §3 related results are proved for κ-like models (κ any regular cardinal) which need not be well founded. As an application a soft proof is given of a theorem of Schmerl on the model-theoretic relation κ → λ. (The author has been informed that Silver had earlier, independently, found a similar unpublished proof of that theorem.) Also, a simpler proof is given of (a generalization of) a characterization by Keisler and Silver of the class of well-founded models which have a Σn end extension for each n ∈ ω. The case κ = ω1 is investigated more deeply in §4, where the problem solved by Theorem 1.1 is considered for non-well-founded models. In Theorems 4.1 and 4.4, ω1-like models of ZFC are constructed which have a Σn end extension for all n ∈ ω but have no elementary end extension. ω1-like models of ZFC which have no Σ3 end extension are produced in Theorem 4.2. The proof uses a notion of satisfaction class, which is also applied in the proof of Theorem 4.6: No model of ZFC has a definable end extension which satisfies ZFC. Finally, Theorem 5.1 generalizes results of Keisler and Morley, and Hutchinson, by asserting that every model of ZFC of countable cofinality has a topless elementary end extension. This contrasts with the rest of the paper, which shows that for well-founded models of uncountable cofinality and for κ-like models with κ regular, topless end extensions are much rarer than blunt end extensions.

In this paper the symbolism, definitions and results of my two papers, namely A test for the existence of tautologies according to many-valued truth-tables (hereafter referred to as ET) and Note on truth-tables (hereafter referred to as TT) will be presupposed.

The problem whether two arbitrary truth-tables with finite number of elements are equal or not was reduced in TT to the question of equality of two identical tables in which different elements have been designated. However, no general method for testing the equality of truth-tables was given there, although some cases were discussed for which such a method is available. In the present paper we shall describe a general method using the considerations of ET and TT.

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.

We work throughout in a finite relational language L. This paper is built on [2] and [3]. We repeat some of the basic notions and results from these papers for the convenience of the reader but familiarity with the setup in the first few sections of [3] is needed to read this paper. Spencer and Shelah [6] constructed for each irrational α between 0 and 1 the theory Tα as the almost sure theory of random graphs with edge probability n−α. In [2] we proved that this was the same theory as the theory Tα built by constructing a generic model in [3]. In this paper we explore some of the more subtle model theoretic properties of this theory. We show that Tα has the dimensional order property and does not have the finite cover property.

We work in the framework of [3] so probability theory is not needed in this paper. This choice allows us to consider a wider class of theories than just the Tα. The basic facts cited from [3] were due to Hrushovski [4]; a full bibliography is in [3]. For general background in stability theory see [1] or [5].

We work at three levels of generality. The first is given by an axiomatic framework in Context 1.10. Section 2 is carried out in this generality. The main family of examples for this context is described in Example 1.3. Sections 3 and 4 depend on a function δ assigning a real number to each finite L-structure as in these examples. Some of the constructions in Section 3 (labeled at the time) use heavily the restriction of the class of examples to graphs. The first author acknowledges useful discussions on this paper with Sergei Starchenko.

A topological classification scheme consists of two ingredients: (1) an abstract class of topological spaces; and (2) a “taxonomy”, i.e. a list of first order sentences, together with a way of assigning an abstract class of spaces to each sentence of the list so that logically equivalent sentences are assigned the same class. is then endowed with an equivalence relation, two spaces belonging to the same equivalence class if and only if they lie in the same classes prescribed by the taxonomy. A space X in is characterized within the classification scheme if whenever Y ∊ and Y is equivalent to X, then Y is homeomorphic to X. As prime example, the closed set taxonomy assigns to each sentence in the first order language of bounded lattices the class of topological spaces whose lattices of closed sets satisfy that sentence. It turns out that every compact two-complex is characterized via this taxonomy in the class of metrizable spaces, but that no infinite discrete space is so characterized. We investigate various natural classification schemes, compare them, and look into the question of which spaces can and cannot be characterized within them.

A computer handles λ-terms more easily if these are translated into combinatory terms. This translation process is called bracket abstraction. The simplest abstraction algorithm—the (fab) algorithm of Curry (see Curry and Feys [6])—is lengthy to implement and produces combinatory terms that increase rapidly in length as the number of variables to be abstracted increases.

There are several ways in which these problems can be alleviated:

(1) A change in order of the clauses in the algorithm so that (f) is performed as a last resort.

(2) The use of an extra clause (c), appropriate to βη reduction.

(3) The introduction of a finite number of extra combinators.

The original 1924 form of bracket abstraction of Schönfinkel [17], which in fact predates λ-calculus, uses all three of these techniques; all are also mentioned in Curry and Feys [6].

A technique employed by many computing scientists (Turner [20], Peyton Jones [16], Oberhauser [15]) is to use the (fab) algorithm followed by certain “optimizations” or simplifications involving extra combinators and sometimes special cases of (c).

Another is either to allow a fixed infinite set of (super-) combinators (Abdali [1], Kennaway and Sleep [10], Krishnamurthy [12], Tonino [19]) or to allow new combinators to be defined one by one during the abstraction process (Hughes [7] and [8]).

A final method encodes the variables to be abstracted as an n-tuple—this requires only a finite number of combinators (Curien [5], Statman [18]).

We present a framework for studying the concept of independence in a general context covering database theory, algebra and model theory as special cases. We show that well-known axioms and rules of independence for making inferences concerning basic atomic independence statements are complete with respect to a variety of semantics. Our results show that the uses of independence concepts in as different areas as database theory, algebra, and model theory, can be completely characterized by the same axioms. We also consider concepts related to independence, such as dependence.

A model is said to be Leibnizian if it has no pair of indiscernibles. Mycielski has shown that there is a first order axiom LM (the Leibniz-Mycielski axiom) such that for any completion T of Zermelo-Fraenkel set theory ZF. T has a Leibnizian model if and only if T proves LM. Here we prove:

Theorem A. Every complete theory T extending ZF + LM has nonisomorphic countable Leibnizian models.

Theorem B. If κ is a prescribed definable infinite cardinal ofa complete theory T extending ZF + V = OD, then there are nonisomorphic Leibnizian models of T of power ℵ1such thatis ℵ1-like.

Theorem C. Every complete theory T extendingZF + V = ODhas nonisomorphic ℵ1-like Leibnizian models.

We introduce a notion of the space of types in positive model theory based on Stone duality for distributive lattices. We show that this space closely mirrors the Stone space of types in the full first-order model theory with negation (Tarskian model theory). We use this to generalise some classical results on countable models from the Tarskian setting to positive model theory.

Assuming CH. Hindman [2] showed that the existence of certain ultrafilters on the power set of the natural numbers is equivalent to Hindman's Theorem. Adapting this work to a countable setting formalized in RCA0, this article proves the equivalence of the existence of certain ultrafilters on countable Boolean algebras and an iterated form of Hindman's Theorem, which is closely related to Milliken's Theorem. A computable restriction of Hindman's Theorem follows as a corollary.

By using a provability predicate of PA, we define ThmPA(M) as the set of theorems of PA in a model M of PA. We say a model M of PA is (1) illusory if ThmPA(M) ⊈ ThmPA(ℕ), (2) heterodox if ThmPA(M) ⊈ TA, (3) sane if M ⊨ ConPA, and insane if it is not sane, (4) maximally sane if it is sane and ThmPA(M) ⊆ ThmPA(N) implies ThmPA(M) = ThmPA(N) for every sane model N of PA. We firstly show that M is heterodox if and only if it is illusory, and that ThmPA(M) ∩ TA ≠ ThmPA(ℕ) for any illusory model M. Then we show that there exists a maximally sane model, every maximally sane model satisfies ¬ConPA+ConPA, and there exists a sane model of ¬ConPA+ConPA which is not maximally sane. We define that an insane model is (5) illusory by nature if its every initial segment being a nonstandard model of PA is illusory, and (6) going insane suddenly if its every initial segment being a sane model of PA is not illusory. We show that there exists a model of PA which is illusory by nature, and we prove the existence of a model of PA which is going insane suddenly.

In [2], Prikry showed that if κ is a weakly inaccessible cardinal which carries a Rowbottom filter, then there is a Boolean extension of V (the universe), having the same cardinals as V, in which cf(κ) = ω. In this note, we obtain necessary and sufficient conditions which a filter D on κ must possess in order that this may be done.

A notation for an ordinal using patterns of resemblance is based on choosing an isominimal set of ordinals containing the given ordinal. There are many choices for this set meaning that notations are far from unique. We establish that among all such isominimal sets there is one which is smallest under inclusion thus providing an appropriate notion of normal form notation in this context. In addition, we calculate the elements of this isominimal set using standard notations based on collapsing functions. This provides a capstone to the results in [2, 6, 8, 9, 7], using further refinement of ordinal arithmetic developed in [8] which then both allows for a simple characterization of normal forms for patterns of order one and will play a key role in the arithmetical analysis of pure patterns of order two, [5].

In a recent paper [3] H. Gaifman investigated some model theoretic consequences of Matijasevič's theorem [5], and posed some further problems which naturally arise. We provide here partial answers to two of these problems, the results having been previously announced in the postscript of [3].

Firstly, it is shown in [3] that if M1 and M2 are models of the Peano axioms P and M1 ⊆ M2, then M1 is closed under the recursive functions of M2. The converse of this statement is false. Moreover, Gaifman asks: Is every initial segment of a model M of P which is closed under the recursive functions of M (or the ∑n-definable functions) also a model of P? We show that this is false and our method gives, en route, another proof of a theorem of Rabin [7] stating the P is not implied by any consistent set of ∑n sentences for any n.

Secondly, we partially answer a question posed on p. 129 of [3] by proving (some-what more than) every countable nonstandard model of P has an end extension in which a diophantine equation, not solvable in the original model, has a solution. We can, in fact, take the new model to be isomorphic to the original one. This generalises (apart from the countability restriction) a theorem of Rabin [6].

Tarski [19] proved the important theorem that the class of representable relation algebras is equationally axiomatizable. One of the key steps in his proof is showing that the class of (isomorphs of) simple set relation algebras—that is, algebras of binary relations with a unit of the form U × U for some non-empty set U —is universal, i.e., is axiomatizable by a set of universal sentences. In the same paper Tarski observed that the class of (isomorphs of) relation algebras constructed from groups (so-called group relation algebras) is also universal.

We shall abstract the essential ingredients of Tarski's method (in Corollary 2.4), and then combine them with some observations about atom structures, to establish (in Theorem 2.6) a rather general method for showing that certain classes of simple relation algebras—and, more generally, certain classes of simple algebras in a discriminator variety V—are universal, and consequently that the collections of (isomorphs of) subdirect products of algebras in such classes form subvarieties of V. As applications of the method we show that two well-known classes of simple relation algebras, those constructed from projective geometries (sometimes called Lyndon algebras) and those constructed from modular lattices with a zero (sometimes called Maddux algebras), are universal. In the process we prove that these two classes consist precisely of all (isomorphs of) complex algebras over the respective geometries and modular lattices, provided that we choose the primitive notions of the latter structures in an appropriate fashion. We also derive Tarski's theorems and a related theorem of the author as easy corollaries of Theorem 2.6.

We study the generic theory of algebraically closed fields of fixed positive characteristic with a predicate for an additive subgroup, called $\mathrm {ACFG}$ . This theory was introduced in [16] as a new example of $\mathrm {NSOP}_{1}$ nonsimple theory. In this paper we describe more features of $\mathrm {ACFG}$ , such as imaginaries. We also study various independence relations in $\mathrm {ACFG}$ , such as Kim-independence or forking independence, and describe interactions between them.

Assume (ℵ0, ℵ1) → (λ, λ+). Assume M is a model of a first order theory T of cardinality at most λ+ in a language of cardinality ≤ λ. Let N be a model with the same language. Let Δ be a set of first order formulas in and let D be a regular filter on λ. Then M is Δ-embeddable into the reduced power Nλ/D, provided that every Δ-existential formula true in M is true also in N. We obtain the following corollary: for M as above and D a regular ultrafilter over λ, Mλ/D is λ++-universal. Our second result is as follows: For i < μ let Mi, and Ni, be elementarily equivalent models of a language which has cardinality ≤ λ. Suppose D is a regular filter on λ and (ℵ0, ℵ1) → (λ, λ+) holds. We show that then the second player has a winning strategy in the Ehrenfeucht-Fraïssé game of length λ+ on ΠiMi/D and ΠiNi/D. This yields the following corollary: Assume GCH and λ regular (or just (ℵ0, ℵ1) → (λ, λ+) and 2λ = λ+. For L, Mi and Ni be as above, if D is a regular filter on λ, then ΠiMi/D ≅ ΠiNi/D.

There has been increasing interest in intuitionistic methods over the years. Still, there has been relatively little work on intuitionistic set theory, and most of that has been on intuitionistic ZF. This investigation is about intuitionistic admissibility and theories of similar strength.

There are several more particular goals for this paper. One is just to get some more Kripke models of various set theories out there. Those papers that have dealt with IZF usually were more proof-theoretic in nature, and did not provide models. Furthermore, the inspirations for many of the constructions here are classical forcing arguments. Although the correspondence between the forcing and the Kripke constructions is not made tight, the relationship between these two methods is of interest (see [6] for instance) and some examples, even if only suggestive, should help us better understand the relationship between forcing and Kripke constructions. Along different lines, the subject of least and greatest fixed points of inductive definitions, while of interest to computer scientists, has yet to be studied constructively, and probably holds some surprises. Admissibility is of course the proper set-theoretic context for this study. Finally, while most of the classical material referred to here has long been standard, some of it has not been well codified and may even be unknown, so along the way we'll even fill in a gap in the classical literature.

The next section develops the basics of IKP, including some remarks on fixed points of inductive definitions.