We prove the hereditary undecidability of the Lt theories of:

(1) torsion-free Hausdorff topological abelian groups;

(2) locally pure Hausdorff topological abelian groups.

In a recent article in this Journal (see [3]), J.P. Jones states and proves a theorem which purports to give an “absolute epistemological upper bound on the complexity of mathematical proofs” for recursively axiomatizable theories. However, Jones' statement of this result is misleading, and in fact defective, as can be seen by a close analysis of it. Such an analysis is the object of the present note.

The main point is that Jones' “epistemological bound” can in no way be considered a computational bound on the complexity of proofs. Not only is the “proof-theoretic interpretation of the number 243” contained in Jones' article objectionable but, more fundamentally, there is in a strong sense no way one can hope to recover anything like the full force suggested by Jones' original statement of the theorem.

We wish to insist that our comments concern only the difficulties surrounding Jones' Corollary 1 on p. 338 of his article and not his ingenious construction of universal Diophantine representations of r.e. sets presented in the same article.

The double nature, both logical and geometrical, of topos theory is one of the most fascinating aspects of this discipline. Consequently, it might be of some interest that an essentially “geometric” fact such as the associated sheaf theorem admits a proof entirely based on methods of categorical logic.

The idea of this proof comes from a previous paper [V], where the same technique was used in the context of Grothendieck topoi. That paper used a generalized notion of forcing which leads directly from classical Tarski semantics in sets to Kripke-Joyal semantics in an arbitrary Grothendieck topos and gives a precise description of the links between the two. On account of this very close relationship, we thus could establish various basic facts on Grothendieck topoi without an extensive use of categorical methods, simply by viewing given subobjects as interpretations of appropriate formulas written in the formal language that had been used all along.

L'absence ou la présence de la propriété d'indépendance, introduite par Shelah, est sans aucun doute une mesure significative de la complexité, du point de vue de la théorie des modèles, d'une théorie instable. Il importe donc de distinguer les théories qui ont cette propriété de celles qui ne l'ont pas; le critère de Keisler [1] et de Shelah [5], qui consiste à compter le nombre de types qu'on peut obtenir sur un ensemble de paramètres de cardinal λ, n'emporte la décision que si l'on nie assez brutalement l'hypothèse du continu généralisée. Je propose ici de compter, étant donnée une partie X, de cardinal λ, d'un espace de types Si(M), quel peut être le cardinal de l'adhérence de X; ce point de vue équivaut au précédent si la théorie T est stable (voir la remarque après le Théorème 7); et si T n'a pas la propriété d'indépendance ce cardinal est au plus 2λ, tandis que si T a la propriété d'indépendance il peut atteindre 22λ (Théorème 7). Cela donne, indépendamment d'hypothèses de théorie des ensembles, une mesure quantitative de la plus grande complexité des espaces de types s'il y a propriété d'indépendance; et sous la forme du Théorème 8, cela devient un test très maniable dans les situations concrètes, car l'expérience m'a prouvé que les cohéritiers d'un type sont toujours faciles à déterminer.

Dans une première section, je considère un type complet p au dessus d'un modèle M d'une théorie complète T, et parmi les fils de p sur une extension élémentaire de M j'en distingue certains que j'appelle fils spéciaux de p: cette notion est naturelle et utile pour la suite, car le critère que j'ai décrit revient à prouver l'abondance de certains fils spéciaux d'un même type; on notera au passage le Théorème 3.

Le théorème de caractérisation de la propriété d'indépendance est prouvé dans une deuxième section, grâce à un résultat de combinatoire (Théorème 6); j'y montre aussi, à titre d'illustration, qu'un ordre total n'a jamais la propriété d'indépendance.

The aim of this paper is to provide an addendum to a paper by Rose with the same title which has appeared in an earlier issue of this Journal [2]. Our new result is:

Theorem. A ring of characteristic zero which admits elimination of quantifiers in the language (0, 1, +, ·) is an algebraically closed field.

Through the ability of arithmetic to partially define truth and the ability of infinite integers to simulate limit processes, nonstandard models of arithmetic automatically have a certain amount of saturation: Any encodable partial type whose formulae all fall into the domain of applicability of a truth definition must, by finite satisfiability and Overspill, be nonstandard-finitely satisfiable—whence realized. This fact was first exploited by A. Robinson, who in Robinson [1963] cited the unrealizability in a given model of a certain encodable partial type to prove Tarski's Theorem on the Undefinability of Truth. A decade later, H. Friedman brought this phenomenon to the public's attention by using it to establish impressive embeddability criteria for countable nonstandard models of arithmetic. Subsequently, Wilkie considered models expandable to “strong theories” and, among such models, complemented Friedman's embeddability criteria with elementary embeddability and isomorphism criteria. Oddly enough, the fact that some kind of saturation property was being employed was not explicitly acknowledged in any of this work.

It is in the unpublished dissertation of Wilmers that these submerged saturation properties first surfaced. [As I have only seen accounts of it (most notably Murawski [1976/1977]) and not the dissertation itself, what I have to say about it will not quite be accurate. (Indeed, the referee has refuted, without providing alternate information, every conjecture I have made about the contents of this thesis.)

It is probably because intuitionism is founded on the concept of (abstract) proof that it has been possible to develop various kinds of models. The following is but a partial list: Gabbay [5], Beth [2], Kripke [8], Kleene [7], Läuchli [9], McKinsey and Tarski [10], Rasiowa and Sikorski [14], Scott [15], de Swart [16], and Veldman [17].

The original purpose for having the models appears to have been for obtaining independence or consistency results for certain formalizations of intuitionism [see Beth [2], Prawitz [13]]; of course, if the models could be also justified as being plausible interpretations of intuitionistic thinking, so much the better. In fact, having some kind of plausible interpretation makes it much easier to work with the models. Occasionally the models were used to suggest possible extensions of the formal systems; for example, the Kripke models with constant domains have motivated interest in the formal logic CD which extends the Intuitionistic Predicate Calculus (IPC) by having the axiom schema

We prove the following algebraic characterization of elementary equivalence: ≡ restricted to countable structures of finite type is minimal among the equivalence relations, other than isomorphism, which are preserved under reduct and renaming and which have the Robinson property; the latter is a faithful adaptation for equivalence relations of the familiar model theoretical notion. We apply this result to Friedman's fourth problem by proving that if is an (ω1, ω)-compact logic satisfying both the Robinson consistency theorem on countable structures of finite type and the Löwenheim-Skolem theorem for some λ < ωω for theories having ω1 many sentences, then ≡L = ≡ on such structures.

Roughly speaking, a pattern is a finite sequence coding the set of natural numbers n for which the Σn+1 projectum is less than the Σn projectum for a given admissible ordinal. We prove that for each pattern there exists an ordinal realizing it. Several results on the orderings of patterns are given. We conclude the paper with remarks on Δn projecta. The main technique, used throughout the paper, is Jensen's Uniformisation Theorem.

This paper continues the study of κ-ultrafilters over a measurable cardinal κ, following the sequence of papers Ketonen [2], Kanamori [1] and Menas [4]. Much of the concern will be with p-point κ-ultrafilters, which have become a focus of attention because they epitomize situations of further complexity beyond the better understood cases, normal and product κ-ultrafilters.

For any κ-ultrafilter D, let iD: V → MD ≃ Vκ/D be the elementary embedding of the universe into the transitization of the ultrapower by D. Situations of U < RKD will be exhibited when iU(κ) < iD(κ), and when iU(κ) = iD(κ). The main result will then be that if the latter case obtains, then there is an inner model with two measurable cardinals. (As will be pointed out, this formulation is due to Kunen, and improves on an earlier version of the author.) Incidentally, a similar conclusion will also follow from the assertion that there is an ascending Rudin-Keisler chain of κ-ultrafilters of length ω + 1. The interest in these results lies in the derivability of a substantial large cardinal assertion from plausible hypotheses on κ-ultrafilters.

Let p be an odd prime. A method is described which given a structure M of finite similarity type produces a nilpotent group of class 2 and exponent p which is in the same stability class as M.

Theorem. There are nilpotent groups of class 2 and exponent p in all stability classes.

Theorem. The problem of characterizing a stability class is equivalent to characterizing the (nilpotent, class 2, exponent p) groups in that class.

The modal logic GL has been found by Solovay [13] to formalize the provable propositional properties of the provability-predicate for Peano's Arithmetic PA (cf. §1 below). We give several sequential calculi for GL, compare their merits, and use one calculus to syntactically derive several metamathematical results about GL.

Some of our results have been proved model theoretically, and similar proofs are fairly straightforward for several of the remaining ones (G. Boolos and the referee have provided such proofs for 4.1, 4.3 and 5.1 below). However, our syntactic techniques often yield more concise and obviously constructive proofs, they offer additional insight into the nature of the systems considered, and are easily adaptable to systems for which semantical analysis is problematic.

I am indebted to G. Boolos and to the referee for their valuable advice. The referee has suggested the rule GL of §3 below as an axiomatization of GL; the resulting sequential calculus has allowed a definite improvement of our original presentation.

It will be shown that system Q of [1], with minor corrections, is free from contradiction and can be strengthened in various ways without destroying its consistency.

By the system C*A will be meant the system that results in place of CA if rule 3.4.10 is omitted in [2] from the rules 3.4 defining PA and RA for an arbitrary class A of equations. We let P*A be the theorems of C*A and R*A be the antitheorems of C*A, just as PA and RA are respectively the theorems and antitheorems of CA in [2]. Similarly Γ* is the class that is defined exactly like the class Γ in 4.18 of [2] except that systems of the form C*A replace those of the form CA. The terminology and conventions of [2] will be assumed in what follows.

Using the methods that were used to show in 5.9 of [2] that CΓ is free from contradiction, it is easy to show that C*A is free from contradiction for every class A of equations. (The fact that rule 3.4.10 is omitted causes the proof of consistency to work even for cases where A is not systematically closed in the sense of 4.11 of [2].)

Just as it was shown in 4.19 of [2] that Γ is systematically closed (in the sense of 4.11 of [2]), so also it clearly can be shown that Γ* is systematically closed. This means, in effect, that the rule of substitution of equals for equals holds in the system C*Γ*.

The model companions of the theories of n-dimensional partial orderings and n-dimensional distributive lattices are found for each finite n. Each model companion is given as the theory of a structure which is specified. The model companions are model completions only for n = 1. The structure of the model companion of the theory of n-dimensional partial orderings is a lattice only for n = 1. Each of the model companions is seen to be finitely axiomatizable, and a set of basic formulas, each of which is existential, is specified for each model companion. Finally a topolo-gically natural notion of dense n-dimensional partial ordering is introduced and shown to have a finitely axiomatizable undecidable theory.

In this paragraph we shall define the notion of model companion (cf. [4]) and indicate the way in which we shall demonstrate that one theory is the model companion of another in this paper. For T and T* theories in a common language, T* is called a model companion of T if and only if the following two conditions are satisfied: first, Tand T* are mutually model consistent, which means that every model of either is embeddable in some model of the other; secondly, T* is model complete, which means that if and are both models of T* and is a substructure of , then is an elementary substructure of . A definition of model completion may be obtained by strengthening the notion of model companion to also require that T* admit elimination of quantifiers. In all of our examples the model companion will have only one countable model. Although the ℵ0-categoricity of the model companions follows from Saracino [8], we give specific proofs since these proofs fit so naturally in our analyses.

This paper deals primarily with countable, simple, connected graphs and the following two conditions which are trivially satisfied if the graphs are finite:

(a) there is an edge-recognition algorithm, i.e., an effective procedure which enables us, given two distinct vertices, to decide whether they are adjacent,

(b) there is a shortest path algorithm, i.e., an effective procedure which enables us, given two distinct vertices, to find a minimal path joining them.

A graph G = ‹ν, η› with ν as set of vertices and η as set of edges is an α-graph if it satisfies (a); it is an ω-graph if it satisfies (b). G is called r.e. (isolic) if the sets ν and η are r.e. (isolated). While every ω-graph is an α-graph, the converse is false, even if G is r.e. or isolic. Several basic properties of finite graphs do not generalize to ω-graphs. For example, an ω-tree with more than one vertex need not have two pendant vertices, but may have only one or none, since it may be a 1-way or 2-way infinite path. Nevertheless, some elementary propositions for finite graphs G = ‹ν, η› with n = card(ν), e = card(η), do generalize to isolic ω-graphs, e.g., n − 1 ≤ e ≤ n(n − 1)/2. Let N and E be the recursive equivalence types of ν and η respectively. Then we have for an isolic α-tree G = ‹ν, η›: N = E + 1 iff G is an ω-tree. The theorem that every finite graph has a spanning tree has a natural, effective analogue for ω-graphs.

The structural difference between isolic α-graphs and isolic ω-graphs will be illustrated by: (i) every infinite graph is isomorphic to some isolic α-graph; (ii) there is an infinite graph which is not isomorphic to any isolic ω-graph.

An isolic α-graph is also called a twilight graph. There are c such graphs, c denoting the cardinality of the continuum.

T. Jech and K. Prikry introduced the concept of precipitous ideals as a counterpart of measurable cardinals for small cardinals (Jech and Prikry [1]). To get a precipitous ideal on ℵ1( W. Mitchell used the Cohen extension by the standard Levy collapse that makes κ = ℵ1, where κ is a measurable cardinal in the ground model (Jech, Magidor, Mitchell and Prikry [2]). His proof essentially used the fact that , where is the notion of forcing of Levy collapse and j is the elementary embedding obtained by a normal ultrafilter on κ.

On the other hand, we know that has the κ-chain condition. In this paper, we show that the κ-chain condition for notions of forcing plays an essential role for preserving normal precipitous ideals. Namely,

Theorem 1. Let κ be a regular uncountable cardinal and I a normal ideal on κ. Let be a notion of forcing with the κ-chain condition. Then, I is precipitous iff ⊩“the ideal on κ generated by I is precipitous”.

As an application of Theorem 1, we have the following theorem.

Theorem 2. Let κ be a regular uncountable cardinal, and Γ be a κ-saturated normal ideal on κ. Then, {a < κ; the ideal of thin sets on a is precipitous} has either Γ-measure one or Γ-measure zero, and the ideal of thin sets on κ is precipitous iff {α< κ; the ideal of thin sets on α is precipitous) has Γ-measure one.

In this paper I will develop a semantic account for modal logic by considering only the values of sentences (and formulas). This account makes no use of possible worlds. To develop such an account, we must recognize four values. These are obtained by subdividing (plain) truth into necessary truth (T) and contingent truth (t); and by subdividing falsity into contingent falsity (f) and necessary falsity (impossibility: F). The semantic account results from reflecting on these concepts and on the meanings of the logical operators.

To begin with, we shall consider the propositional language L0. The language L0 has (1) infinitely many atomic sentences, (2) the two truth-functional connectives ∼, ∨, and the modal operator □. (Square brackets are used for punctuation.) The other logical expressions are defined as follows:

D1 [A & B] = (def)∼[∼A ∨ ∼ B],

D2 [A ⊃ B] = (def)[∼A ∨ B],

D3 ◊ A =(def)∼□∼A.

I shall use matrices to give partial characterizations of the significance of logical expressions in L0. For negation, this matrix is wholly adequate:

Upon reflection, it should be clear that this matrix is the obviously correct matrix for negation.

A Boolean algebra is recursive if B is a recursive subset of the natural numbers N and the operations ∧ (meet), ∨ (join), and ¬ (complement) are partial recursive. Given two Boolean algebras and , we write if is isomorphic to and if is recursively isomorphic to , that is, if there is a partial recursive function f: B1 → B2 which is an isomorphism from to . will denote the set of atoms of and () will denote the ideal generated by the atoms of .

One of the main questions which motivated this paper is “To what extent does the classical isomorphism type of a recursive Boolean algebra restrict the possible recursion theoretic properties of ?” For example, it is easy to see that must be co-r.e. (i.e., N − is an r.e. set), but can be immune, not immune, cohesive, etc? It follows from a result of Goncharov [4] that there exist classical isomorphism types which contain recursive Boolean algebras but do not contain any recursive Boolean algebras such that is recursive. Thus the classical isomorphism can restrict the possible Turing degrees of , but what is the extent of this restriction? Another main question is “What is the recursion theoretic relationship between and () in a recursive Boolean algebra?” In our attempt to answer these questions, we were led to a wide variety of recursive isomorphism types which are contained in the classical isomorphism type of any recursive Boolean algebra with an infinite set of atoms.

Unless otherwise stated, every structure in this paper is countable in a countable, actually recursive language and every formula is one of .

The definition of the so-called canonical Scott-sentence of a structure M, CSS(M) (compare Nadel [8]), is based on the ordinal invariant called the Scott-height of M, denoted SH(M) (compare Makkai [5]). To describe SH(M), let “b ≡αα”, for finite tuples of equal lengths b and a of elements of M and any ordinal α, stand for “b and a satisfy (in M) the same formulas of quantifier-rank ≤ α” (for the quantifier rank of a formula, see e.g., Barwise [1]); also, let “b ∼ a” mean that there is an automorphism of M mapping b to a. With a fixed M, and a in M, sh(a) (or shM(a)) denotes the least ordinal α such that for all b in M, b ≡αa implies b ∼ a. SH(M) is the least ordinal α such that for all a and b in M, b ≡αa implies b ∼ a; hence SH(M) = sup{sh(a): a in M}.

This paper is a sequel to [3] and it contains, among other things, proofs of the results announced in the last section of that paper.

In §1, we use the general method of [3] together with reflection arguments to study the properties of forcing with Δ perfect trees, for certain Spector pointclasses Γ, obtaining as a main result the existence of a continuum of minimal Δ-degrees for such Γ's, under determinacy hypotheses. In particular, using PD, we prove the existence of continuum many minimal Δ½n+1-degrees, for all n.

Following an idea of Leo Harrington, we extend these results in §2 to show the existence of minimal strict upper bounds for sequences of Δ-degrees which are not too far apart. As a corollary, it is computed that the length of the natural hierarchy of Δ½n+1-degrees is equal to ω when n ≥ 1. (By results of Sacks and Richter the length of the natural hierarchy of -degrees is known to be equal to the first recursively inaccessible ordinal.)

We will follow in this paper standard notation and terminology, as in Moschovakis' book [7]. Letters e, i, j, k, l, m, n vary over the set of natural numbers ω, a, b, c over the Cantor space 2ω and α, β, γ, δ, … over the set of reals ωω. Finally ξ, η, κ, λ always denote ordinals.