We study large cardinal properties associated with Ramseyness in which homogeneous sets are demanded to satisfy various transfinite degrees of indescribability. Sharpe and Welch [25], and independently Bagaria [1], extended the notion of $\Pi ^1_n$ -indescribability where $n<\omega $ to that of $\Pi ^1_\xi $ -indescribability where $\xi \geq \omega $ . By iterating Feng’s Ramsey operator [12] on the various $\Pi ^1_\xi $ -indescribability ideals, we obtain new large cardinal hierarchies and corresponding nonlinear increasing hierarchies of normal ideals. We provide a complete account of the containment relationships between the resulting ideals and show that the corresponding large cardinal properties yield a strict linear refinement of Feng’s original Ramsey hierarchy. We isolate Ramsey properties which provide strictly increasing hierarchies between Feng’s $\Pi _\alpha $ -Ramsey and $\Pi _{\alpha +1}$ -Ramsey cardinals for all odd $\alpha <\omega $ and for all $\omega \leq \alpha <\kappa $ . We also show that, given any ordinals $\beta _0,\beta _1<\kappa $ the increasing chains of ideals obtained by iterating the Ramsey operator on the $\Pi ^1_{\beta _0}$ -indescribability ideal and the $\Pi ^1_{\beta _1}$ -indescribability ideal respectively, are eventually equal; moreover, we identify the least degree of Ramseyness at which this equality occurs. As an application of our results we show that one can characterize our new large cardinal notions and the corresponding ideals in terms of generic elementary embeddings; as a special case this yields generic embedding characterizations of $\Pi ^1_\xi $ -indescribability and Ramseyness.

Problems of computing probabilities of statements about large, finite structures have become an important subject area of finite combinatorics. Within the last two decades many researchers have turned their attention to such problems and have developed a variety of methods for dealing with them. Applications of these ideas include nonconstructive existence proofs for graphs with certain properties by showing the properties occur with nonzero probabilities (for examples see Erdös and Spencer [ES] and Bollobás [Bo]), and determination of average running times for sorting algorithms by computing asymptotic probabilities of statements about permutations (see Knuth [Kn]). Two types of techniques recur in solutions to such problems: probabilistic techniques, such as those used by Erdös and Spencer [ES], and classical assymptotic techniques, such as those surveyed by Bender [Be] and de Bruijn [Br]. Studying this body of techniques, one notices that characteristics of these problems suggest certain methods of solution, in much the same way that the form of an integrand may suggest certain substitutions. The question arises, then, as to whether there is a systematic way to approach these problems: is there an algorithm for computing asymptotic probabilities? I will show that the answer is “no”—for any reasonable formulation, the problem of computing asymptotic probabilities is undecidable.

The main theorem of the paper is Theorem 1.6, which says that there is a finitely axiomatizable class in which every first order sentence has an asymptotic probability of 0 or 1—i.e., is almost always true or almost always false in finite structures—but for which the problem of deciding whether a sentence has asymptotic probability 0 or 1 is undecidable. Heretofore, classes known to have such a 0-1 law have had decidable asymptotic probability problems (see Lynch [Ly] for examples and a discussion of previous work in the area).

In a recent paper Professor Wesley Salmon1 has shown that a certain class of c-functions, including the c-function c*. described by Rudolph Carnap2 yield incompatible results when applied to the same sentences in two languages systems which, though they have the same individual constants, do not have the same predicates. Each c-function of the class in question is characterized by a parameter λ which is a function of the number of Q-predicates in the language system in which the c-function is used.3 Taking c* as representative of this class of c-functions, I shall argue that Professor Salmon's results do not provide a reasonable basis for rejecting such c-functions in favor of others. More specifically, I shall argue (i) that such c-functions yield incompatible results in two languages because not both of the languages are sufficiently complete, (ii) that for any two languages in which such c-functions yield incompatible results there is a rule that will select either the more complete of the two languages or a language that is more complete than either of the two languages, and (iii) that it is impossible for such c-functions to yield incompatible results in two languages that are equally complete.

In 2007, Terence Tao wrote on his blog an essay about soft analysis, hard analysis and the finitization of soft analysis statements into hard analysis statements. One of his main examples was a quasi-finitization of the infinite pigeonhole principle IPP, arriving at the “finitary” infinite pigeonhole principle FIPP1. That turned out to not be the proper formulation and so we proposed an alternative version FIPP2. Tao himself formulated yet another version FIPP3 in a revised version of his essay.

We give a counterexample to FIPP1 and discuss for both of the versions FIPP2 and FIPP3 the faithfulness of their respective finitization of IPP by studying the equivalences IPP ↔ FIPP2 and IPP ↔ FIPP3 in the context of reverse mathematics ([9]). In the process of doing this we also introduce a continuous uniform boundedness principle CUB as a formalization of Tao's notion of a correspondence principle and study the strength of this principle and various restrictions thereof in terms of reverse mathematics, i.e., in terms of the “big five” subsystems of second order arithmetic.

Es ist bekannt, daβ man aus den Funktionen 0 und n + 1 ausgehend, die in der elementaren Zahlentheorie benutzten Funktionen durch eine endliche Kette von Substitutionen und “primitiven Rekursionen” der Form

(α(a1, …, ar) und β(n, a1, …, ar, ar+1) bereits eingeführte Funktionen) definieren kann. Aber die “mehrfache Rekursion”, wobei die Rekursion gleichzeitig nach mehreren Variablen läuft, führt aus der so entstehenden Klasse der “primitiv-rekursiven Funktionen” hinaus. Die sogenannten “mehrfachen”, und zwar “k-rekursiven” Funktionen (zu welchen als Spezialfall auch die primitiv-rekursiven Funktionen gehören) lassen sich von gewissen Ausgangsfunktionen ausgehend durch eine endliche Kette von Substitutionen und “k-fachen Rekursionen” (wobei die Rekursion gleichzeitig nach k Variablen läuft) einer gewissen Normalform aufbauen. Die k + 1-fache Rekursion führt für ein jedes k aus der Klasse der k-rekursiven Funktionen hinaus. Das kann mit Hilfe des Diagonalverfahrens bewiesen werden. Das Diagonalverfahren auf sämtliche mehrfach-rekursive Funktionen angewandt (auch wenn man sich auf Funktionen von einer gegebenen Variablenanzahl beschränkt) würde als Beispiel einer nicht mehrfachrekursiven Funktion eine Funktion ergeben, bei der es vom ersten Argument abhängt, wievielstellig die Funktion ist.

Man sagt allgemein, daβ eine Funktion durch irgendeine Rekursion definiert wird, wenn ihr Wert an gewissen Anfangsstellen angegeben, und an einer anderen Stelle aus Werten aufgebaut wird, welche die Funktion an vorherigen Stellen angenommen hat. Es kann aber verschieden vorgeschrieben werden, welche Stellen als Vorgänger einer gegebenen Stelle zu betrachten sind.

A demonstrably consistent theory of real numbers has been outlined by the writer in An extension of basic logic1 (hereafter referred to as EBL). This theory deals with non-negative real numbers, but it could be easily modified to deal with negative real numbers also. It was shown that the theory was adequate for proving a form of the fundamental theorem on least upper bounds and greatest lower bounds. More precisely, the following results were obtained in the terminology of EBL: If С is a class of U-reals and is completely represented in Κ′ and if some U-real is an upper bound of С, then there is a U-real which is a least upper bound of С. If D is a class of (U-reals and is completely represented in Κ′, then there is a U-real which is a greatest lower bound of D.

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.

For a first-order formula φ(x; y) we introduce and study the characteristic sequence (Pn: n < ω) of hypergraphs defined by . We show that combinatorial and classification theoretic properties of the characteristic sequence reflect classification theoretic properties of φ and vice versa. The main results are a characterization of NIP and of simplicity in terms of persistence of configurations in the characteristic sequence. Specifically, we show that some tree properties are detected by the presence of certain combinatorial configurations in the characteristic sequence while other properties such as instability and the independence property manifest themselves in the persistence of complicated configurations under localization.

This paper presents a unified framework that explains and extends the already successful applications of the Leibniz operator, the Suszko operator, and the Tarski operator in recent developments in abstract algebraic logic. To this end, we refine Czelakowski’s notion of an S-compatibility operator, and introduce the notion of coherent family of S-compatibility operators, for a sentential logic S. The notion of coherence is a restricted property of commutativity with inverse images by surjective homomorphisms, which is satisfied by both the Leibniz and the Suszko operators. We generalize several constructions and results already existing for the mentioned operators; in particular, the well-known classes of algebras associated with a logic through each of them, and the notions of full generalized model of a logic and a special kind of S-filters (which generalizes the less-known notion of Leibniz filter). We obtain a General Correspondence Theorem, extending the well-known one from the theory of protoalgebraic logics to arbitrary logics and to more general operators, and strengthening its formulation. We apply the general results to the Leibniz and the Suszko operators, and obtain several characterizations of the main classes of logics in the Leibniz hierarchy by the form of their full generalized models, by old and new properties of the Leibniz operator, and by the behaviour of the Suszko operator. Some of these characterizations complete or extend known ones, for some classes in the hierarchy, thus offering an integrated approach to the Leibniz hierarchy that uncovers some new, nice symmetries.

We study automorphism groups of randomizations of separable structures, with focus on the ℵ0-categorical case. We give a description of the automorphism group of the Borel randomization in terms of the group of the original structure. In the ℵ0-categorical context, this provides a new source of Roelcke precompact Polish groups, and we describe the associated Roelcke compactifications. This allows us also to recover and generalize preservation results of stable and NIP formulas previously established in the literature, via a Banach-theoretic translation. Finally, we study and classify the separable models of the theory of beautiful pairs of randomizations, showing in particular that this theory is never ℵ0-categorical (except in basic cases).

We show here that the first order theory of the positive integers equipped with multiplication remains decidable when one adds to the language the usual order restricted to the prime numbers. We see moreover that the complexity of the latter theory is a tower of exponentials, of height O(n).

We consider the Turing degrees of prime models of complete decidable theories. In particular we show that every complete decidable atomic theory has a prime model whose elementary diagram is low. We combine the construction used in the proof with other constructions to show that complete decidable atomic theories have low prime models with added properties.

If we have a complete decidable atomic theory with all types of the theory computable, we show that for every degree d with 0 < d < 0', there is a prime model with elementary diagram of degree d. Indeed, this is a corollary of the fact that if T is a complete decidable theory and L is a computable set of c.e. partial types of T, then for any degree d > 0, T has a d-decidable model omitting the nonprincipal types listed by L.

We show that from large cardinals it is consistent to have the tree property simultaneously at ${\aleph _{{\omega ^2} + 1}}$ and ${\aleph _{{\omega ^2} + 2}}$ with ${\aleph _{{\omega ^2}}}$ strong limit.

We say that a first order formula Φ distinguishes a structure M over a vocabulary L from another structure M′ over the same vocabulary if Φ is true on M but false on M′. A formula Φ defines an L-structure M if Φ distinguishes M from any other non-isomorphic L-structure M′. A formula Φ identifies an n-element L-structure M if Φ distinguishes M from any other non-isomorphic n-element L-structure M′.

We prove that every n-element structure M is identifiable by a formula with quantifier rank less than and at most one quantifier alternation, where k is the maximum relation arity of M. Moreover, if the automorphism group of M contains no transposition of two elements, the same result holds for definability rather than identification.

The Bernays-Schönfinkel class consists of prenex formulas in which the existential quantifiers all precede the universal quantifiers. We prove that every n-element structure M is identifiable by a formula in the Bernays-Schönfinkel class with less than quantifiers. If in this class of identifying formulas we restrict the number of universal quantifiers to k, then less than quantifiers suffice to identify M and. as long as we keep the number of universal quantifiers bounded by a constant, at total quantifiers are necessary.

We show that the following results in the classical theory of unbounded linear operators on Hilbert spaces can be proved within the framework of Bishop's constructive mathematics: the Kato-Rellich theorem, the spectral theorem. Stone's theorem, and the self-adjointness of the most common quantum mechanical operators, including the Hamiltonians of electro-magnetic fields with some general forms of potentials.

The present note is conceived as a sequel to Mostowki's paper, Über die Unabhängigkeit des Wohlordnungssatzes vom Ordnungsprinzip (which will be quoted as “M.”).

In that paper Mostowski considers a system of axioms , very close to the system of axiomatic set theory of Bernays, but where the axiom of choice is not supposed to hold. Mostowski proves that the well-ordering theorem (Wohlordnungssatz) cannot be derived from the system plus the principle of simple ordering (Ordnungsprinzip).

Recall that a group G has finitely satisfiable generics (fsg) or definable f-generics (dfg) if there is a global type p on G and a small model $M_0$ such that every left translate of p is finitely satisfiable in $M_0$ or definable over $M_0$, respectively. We show that any abelian group definable in a p-adically closed field is an extension of a definably compact fsg definable group by a dfg definable group. We discuss an approach which might prove a similar statement for interpretable abelian groups. In the case where G is an abelian group definable in the standard model $\mathbb {Q}_p$, we show that $G^0 = G^{00}$, and that G is an open subgroup of an algebraic group, up to finite factors. This latter result can be seen as a rough classification of abelian definable groups in $\mathbb {Q}_p$.

We prove that all known examples of weakly o-minimal nonvaluational structures have no definable Skolem functions. We show, however, that such structures eliminate imaginaries up to definable families of cuts. Along the way we give some new examples of weakly o-minimal nonvaluational structures.