The problem of changing the cofinality of a measurable cardinal to ω with the help of an iterated ultrapower construction has been introduced in [Bu] and more completely studied in [De]. The aim of this paper is to investigate how the construction above has to be changed to obtain an uncountable cofinality for the (previously) measurable cardinal.

A forcing approach of this question has been developed by Magidor in [Ma]. Just as in the countable case with Prikry forcing, it turns out that the needed hypothesis and the models constructed are the same in both techniques. However the ultrapowers yield a solution which may appear as more effective. In particular the sequence used to change the measurable cardinal into a cardinal of cofinality α has the property that for any β < α the restriction to β of this sequence can be used to change the cofinality of the (same) measurable cardinal to β.

The result we prove is as follows:

Theorem. Assume that α is a limit ordinal, that (Uβ)β<α is a sequence of complete ultrafilters on κ > α in the model N0, andfor B included in α let NB be the ultrapower of N0 by those Uβ which are such that β is in B.

The permutation method was first applied to NF by Scott; other workers have published results ([2], [3]; or see [4] for a survey). Some of the results proved here have a more metalogical character than most previously yielded by this method. Hinnion and Petry (unpublished, but see [4]) proved that the existence of objects x such that x = {y: x∈y} is consistent with NF. (The significance of this is that the operation that sends x to {y: x∈y} respects ∈ and is thus an embedding.) It is demonstrated below that the existence of such objects is independent of the axioms of NF.

The existence of nontrivial automorphisms of the universe is not an interesting possibility in ZF, since it contradicts wellfoundedness. Similar auguments are not available in NF, however, and it is shown below that if NF + AC for pairs is consistent, then we can consistently add an axiom stating that there is a ∈-automorphism of the universe that is a set of the model. (In the proof given below, the automorphism is in fact of order 2, but natural enrichments of the construction enable one to find automorphisms of other orders with suitable versions of choice as additional hypotheses.)

Le paradigme de théorie stable est la théorie T d'un corps algébriquement clos; une autre théorie stable T′ est celle de la structure formée d'un corps algébriquement clos, avec en outre un symbole relationnel unaire interprétant un de ses sous-corps propres algébriquement clos. C'est à l'éclaircissement des rapports de T et de T′ qu'est consacré cet article.

J'y considère une théorie complète T stable, et les structures formées d'un modèle N de T, avec en outre un symbole relationnel unaire (x) interprétant une restriction élémentaire M de N; j'appelle ces structures paires de modèles de T.

Et je dis que la paire (N, M) est belle si d'une part M est ∣T∣+-saturé, et d'autre part pour tout n-uplet ā d'éléments de N, tout type, au sens de T, sur M ⋃ {α} est réalisé dans N.

Le premier résultat (Théorème 4) est que deux belles paires sont élémentairement équivalentes. Plus précisément, si (N1, M1) et (N2, M2) sont deux belles paires, et si ā est dans la première, b¯ dans la seconde, le fait que le type de ā sur M1 et celui de b¯ sur M2 soient équivalents dans l'ordre fondamental au sens de T suffit (et est bien sûr nécessaire) pour que ā at b¯ aient même type (sur ⊘) au sens de la théorie T′ des belles paires.

The set Λ of isols was extensively studied by Dekker and Myhill in [1]. Subsequently, Nerode [3] developed the theory of Λ(A), the set of isols relative to some recursively closed set of functions A.

One of the main areas of interest of [1] was the natural partial order ≤ on Λ. In this paper we will examine some of the properties of ≤A on Λ(A). We use the following notations: ∣A∣ is the cardinality of the set A, ⊃ denotes strict inclusion, (a) is the power set of the set a, c is the cardinality of the continuum, and ω = {0, 1, 2, …}. The terms A-isol, A-immune, A-r.e., A-incomparable, etc. all refer to the usual meaning of these words, only taken in the context of the recursively closed set A. ReqA(a) is the A-r.e.t. of which a is a representative. By identifying a finite natural number with the A-r.e.t. consisting of sets of a given finite cardinality we see that ω ⊆ Λ(A); Λ(A) is said to be nontrivial iff ω ⊃ Λ(A). The three results proven in this paper are:

1. Theorem 1. If Λ(A) is nontrivial, then ∣Λ(A)∣ = c.

2. Theorem 2. If∣A∣ < c, then Λ(A) is nontrivial.

3. Theorem 3. If ∣A∣ < c and ∣⊿∣ < c and ⊿ ⊆ Λ(A) − ω, then there is aΓ ⊆ Λ(A) − ω such that:

• (a) ∣Γ∣ = c.

• (b) Every member of Γ is A-incomparable with every member of Δ.

• (c) Any two distinct members of Γ are A-incomparable.

We study cofinal extensions of models of arithmetic, in particular we show that some properties near to expandability are preserved under cofinal extensions.

The incompleteness of ZF set theory leads one to look for natural extensions of ZF in which one can prove statements independent of ZF which appear to be “true”. One approach has been to add large cardinal axioms. Or, one can investigate second-order expansions like Kelley-Morse class theory, KM. In this paper we look at a set theory ZF(aa), with an added quantifier aa which ranges over ordinals. The “aa” stands for “almost all”, and although we will consider interpretations in terms of the closed unbounded filter on a regular cardinal κ, we will consider other interpretations also.

We start in §1 by giving the axioms for the theory ZF(aa) and presenting a completeness theorem which gives a model-theoretic definition of ZF(aa). In §2 we investigate set theory with a satisfaction predicate and interpret it in a fragment of ZF(aa). In §3 we generalize the methods of §2 to obtain a hierarchy of satisfaction predicates. We use these predicates to prove reflection theorems, as well as to prove the consistency of certain fragments of ZF(aa). Next, in §4 we discuss expandability of models of ZF to models of fragments of ZF(aa) and of Kelley-Morse. We conclude in §5 with a discussion of an extension ZF(aa) + DET of ZF(aa) in which the quantifier aa is self-dual.

We see far away, Newton said, if we stand on giants' shoulders. We take him seriously here and moreover (as appropriate to recursion-theorists) we will jump from one giant to another, since this paper is mostly an exegesis of two fundamental works: Feferman's Some applications of the notions of forcing and generic sets [4] and Sacks' Forcing with perfect closed sets [19]. We hope the reader is not afraid of heights: our exercises are risky ones, since the two giants are in turn on the shoulders of others! Feferman [4] rests on the basic works of Cohen [2], who introduced forcing with finite conditions in the context of set theory; Sacks [19] relies on Spector [24], who realized—in recursion theory—the necessity of more powerful approximations than the finite ones.

To minimize the risk we will try to keep technicalities to a minimum, choosing to give priority to the methodology of forcing. We do not suppose any previous knowledge of forcing in the reader, but we do require some acquaintance with recursion theory. After all, our interest lies in the applications of the forcing method to the study of various recursion-theoretic notions of degrees. The farther we go, the deeper we plunge into recursion theory.

In Part I only very basic notions and results are used, like the definitions of the arithmetical hierarchy and of the jump operator and their relationships.

This paper is devoted to the study of recursively and short recursively saturated models of PA by means of so-called nonstandard satisfaction methods. The paper is intended to be self-contained. In particular, no knowledge of nonstandard satisfaction classes is assumed. In fact we shall not use this notion explicitly.

We define a certain property of models of PA which we call the S-property and prove that properly short recursively saturated models (see Definition 2.1. below) are exactly short models with the S-property. The main result is that all properly short recursively saturated models are elementary cuts of recursively saturated models. This is a generalization to the uncountable case of the theorem of C. Smorynski [9] and is an easy application of some general results concerning cofinal extensions of models of PA which we discuss in §3.

On the way we obtain another proof of the result of Smorynski and Stavi [10] which says that short recursive and recursive saturation is preserved under cofinal extensions.

The author wants to thank H. Kotlarski and W. Marek for valuable suggestions concerning the subject of the paper.

Special thanks must also go to J. Paris for the lemma used in the proof of Theorem 3.5.

An example is given of a complete theory with minimal models of arbitrarily large minimality rank.

Associated with each first-order theory is a Boolean algebra of sentences and a Boolean space of models. Homomorphisms between the sentence algebras correspond to continuous maps between the model spaces. To what do recursive homomorphisms correspond? We introduce axiomatizable maps as the appropriate dual. For these maps we prove a Cantor-Bernstein theorem. Duality and the Cantor-Bernstein theorem are used to show that the Boolean sentence algebras of any two undecidable languages or of any two functional languages are recursively isomorphic where a language is undecidable iff it has at least one operation or relation symbol of two or more places or at least two unary operation symbols, and a language is functional iff it has exactly one unary operation symbol and all other symbols are for unary relations, constants, or propositions.

Cet article ne contient rien de révolutionnaire, et son auteur a conscience du risque qu'il court de le voir dépassé, lors de sa parution, par des travaux plus profonds. Il a pensé qu'il n'était pas inutile de publier, faute de mieux, les résultats simples auxquels il est parvenu (le lecteur lui saura gré de ce caractère reposant), en espérant qu'il passeront à la postérité sous forme d'exercices dans les manuels futurs où les petits enfants apprendront la théorie des modèles.

Il s'agit de groupes stables; on sait depuis longtemps que la stabilité de la théorie d'un groupe impose des “conditions de chaîne” sur ses sous-groupes définissables : l'exploitation de ce phénomène, de nature bien algébrique, qui est brièvement exposé dans la première section de cet article, a fait le bonheur d'une génération de théoriciens des modèles. Un autre phénomène, plus subtil, semble ne pas avoir épuisé sa substance : c'est celui, dont l'apparition remonte aux travaux de B. Zilber, qui est décrit dans [7] sous le nom de “types de strate maximum”, et dans [2] sous le nom de “large sets”; la deuxième section lui est consacrée: elle a été écrite dans le désir de montrer l'équivalence de ces deux approches, et aussi par repentir de n'avoir pas suffisament éclairé ses motivations dans [7]. Ce souci d'unification a eu une influence, que j'espère salutaire, sur le vocabulaire: je parle ici de “types génériques” et d'“ensembles (ou de formules) génériques”. Ces termes, ainsi que celui de “composante connexe”, sont empruntés au langage de la géométrie: un groupe algébrique, étant définissable dans la théorie d'un corps algébriquement clos, est stable, et ses “types génériques” sont les “points génériques” des géomètres.

We say that a set A of reals is recursive in a real y together with a set B of reals if one can imagine a computing machine with an ability to perform a countably infinite sequence of program steps in finite time and with oracles for B and y so that decides membership in A for any real x input to by way of an oracle for x. We write A ≤ yB. A precise definition of this notion of recursion was first considered in Kleene [9]. In the notation of that paper, A ≤yB if there is an integer e so that χA(x) = {e}(x y, χB, 2E). Here χA is the characteristic function of A. Thus Kleene would say that A is recursive in (y, B, 2E), where 2E is the existential integer quantifier.

Gandy [5] observes that the halting problem for infinitary machines such as , as in the case of Turing machines, gives rise to a jump operator for higher type recursion. Thus given a set B of reals, the superjump B′ of B is defined to be the set of all triples 〈e, x, y〉 such that the eth machine with oracles for y and B eventually halts when given input x. A set A is said to be semirecursive in y together with B if for some integer e, A is the cross section {x: 〈e, x, y 〉 ∈ B′}. In Kleene [9] it is demonstrated that a set A is semirecursive in y alone if and only if it is

This paper looks at linear orders in the following way. A preordering is given, which is linear and recursively enumerable. By performing the natural identification, one obtains a linear order for which equality is not necessarily recursive. A format similar to Metakides and Nerode's [3] is used to study these linear orders. In effective studies of linear orders thus far, the law of antisymmetry (x ≦ y ∧ y ≦ x ⇒ y) has been assumed, so that if the order relation x ≦ y is r.e. then x < y is also r.e. Here the assumption is dropped, so that x < y may not be r.e. and the equality relation may not be recursive; the possibility that equality is not recursive leads to new twists which sometimes lead to negative results.

Reported here are some interesting preliminary results with simple proofs, which are obtained if one looks at these objects with a view to doing recursion theory in the style of Metakides and Nerode. (This style, set in [3], is seen in many subsequent papers by Metakides and Nerode, Kalantari, Remmel, Retzlaff, Shore, and others, e.g. [1], [4], [6], [7], [8], [11]. In a sequel, further investigations will be reported which look at r.e. presented linear orders in this fashion and in the context of Rosenstein's comprehensive work [10].

Obviously, only countable linear orders are under consideration here. For recursion-theoretic notation and terminology see Rogers [9].

The notations and the definitions used here are identical to those in [B.L.]. We also assume that the reader is familiar with the properties of the Rudin-Keisler order on types [L.].

Let T denote an ℵ0-stable countable theory. The main result in [B.L.] is: All countable models of T are almost homogeneous if and only if T satisfies

(*) For all models M of T and all ā ∈ M, if p ∈ S1(ā) is strongly regular multidimensional, then Dim(p; M) ≥ ℵ0.

In this paper we investigate the case of a theory T which does not satisfy condition (*) and, under certain additional assumptions, we construct nonisomorphic and non-almost-homogeneous countable models. The same type of construction has been used before to show that if T is multidimensional, then for all α ≥ 1, T has at least nonisomorphic models of cardinality ℵα [La.], [Sh.]. As a corollary of our main theorem (Theorem 6) and of the previous result in [B.L.], we prove Vaught's Conjecture (and, in fact, Martin's Strong Conjecture) for theories T with αT finite.

Although we are here interested in countable models, we can also note that our construction proves that theories satisfying the assumptions in Theorem 6 have at least nonisomorphic models of cardinality ℵα.

Compactness of supervaluational semantics has long been an open problem. Recently, Peter Woodruff [2] showed that the full quantificational language is not compact. In the present note, I will show that the quantifier-free fragment of the language is compact. Since this result can be easily extended to the monadic quantificational language, and since Woodruff's result only requires the presence of one binary predicate, the two results combined give a complete solution of the problem.

My language L contains infinitely many individual constants, infinitely many n-ary predicates (for every n > 0), the usual connectives, the symbols E! and =, and parentheses. (Atomic) sentences are defined as usual, and Ab/a is the result of uniformly substituting b for a in the sentence A.

An interpretation of L is an ordered pair I = 〈D, φ〉, where D is a set (possibly empty) and φ is a unary function, total from the set of n-ary predicates into the power set of Dn, and partial from the set of constants into D.

A classical valuation for an interpretation I = 〈D, φ〉 is a total unary function V from the set of sentences into {T, F} such that

(a) if φ is defined for all of a1, …, am, then V(Pa1 … an) = T iff 〈φ(a1), …, φ(an)〉 ∈ φ(P);

(b) if φ(a) and φ(b) are both defined, then V(a = b) = T iff φ(a) = φ(b;

(c) if exactly one of φ(a) and φ(b) is defined, then V(a = b) = F;

(d) V(a = a) = T;

(e) if V(a = b) = T and A is atomic, then V(A) = V(Ab/a);

(f)V(E!a) = T iff φ(a) is defined;

(g) V(~ A) = T iff V(A) = F, and similarly for the other connectives.

Assume ZFC + “There is a weakly compact cardinal” is consistent. Then:

(i) For every S ⊆ ω, ZFC + “S and the monadic theory of ω2 are recursive each in the other” is consistent; and

(ii) ZFC + “The full second-order theory of ω2 is interpretable in the monadic theory of ω2” is consistent.

We give here an elementary proof of a recent result of Girard [4] comparing the rates of growth of the two principal (and extreme) examples of a spectrum of “majorization hierarchies”—i.e. hierarchies of increasing number-theoretic functions, indexed by (systems of notations for) initial segments I of the countable ordinals so that if α < β ∈ I then the βth function dominates the αth one at all but finitely-many positive integers x.

Hardy [5] was perhaps the first to make use of a majorization hierarchy—the Hα's below—in “exhibiting” a set of reals with cardinality ℵ1. More recently such hierarchies have played important roles in mathematical logic because they provide natural classifications of recursive functions according to their computational complexity. (All the functions considered here are “honest” in the sense that the size of their values gives a measure of the number of steps needed to compute them.)

The hierarchies we are concerned with fall into three main classes depending on their mode of generation at successor stages, the other crucial parameter being the initial choice of a particular (standard) fundamental sequence λ0 < λ1 < λ2 < … to each limit ordinal λ under consideration which, by a suitable diagonalization, will then determine the generation at stage λ.

Our later comparisons will require the use of a “large” initial segment I of proof-theoretic ordinals, extending as far as the “Howard ordinal”. However we will postpone a precise description of these ordinals and their associated fundamental sequences until later.

The class of 2-square ultrafilters on ω equals the union, for n ≥ 1, of the classes of strictly n Ramsey ultrafilters on ω.

Higher-order theories of properties, relations, and propositions (PRPs) are known to be essentially incomplete relative to their standard notions of validity. There is, however, a first-order theory of PRPs that results when standard first-order logic is supplemented with an operation of intensional abstraction. It turns out that this first-order theory of PRPs is provably complete with respect to its standard notions of validity. The construction involves the development of a new algebraic semantic method. Unlike most other methods used in contemporary intensional logic, this method does not appeal to possible worlds as a heuristic; the heuristic used is that of PRPs taken as primitive entities. This is important, for even though the possible-worlds approach is useful in treating modal logic, it seems to be of little help in treating the logic for psychological matters. The present approach, by contrast, appears to make a step in the direction of a satisfactory treatment of both modal and intentional logic. For, by taking PRPs as primitive entities, we remain free to tailor the statement of their identity conditions so that it agrees with the logical data—modal, psychological, etc. In this way, the present approach suggests a strategy for developing a comprehensive treatment of intensional logic.

In [1] and [2] I explore this prospect philosophically. The purpose of the present paper is to lay out the technical details of the approach and to present the completeness results.

The study of preservation theorems for first order logic was the focus of much research by model theorists in the 1960's. These theorems, which came to form the foundation for classical model theory, characterize first order sentences and theories that are preserved under operations such as the taking of unions or submodels (see Chang and Keisler [5] for a discussion of preservation theorems for first order logic). In current model theoretic research, logics richer than first order logic and applications of logic to other parts of mathematics have assumed the central position. In the former area, preservation theorems are not so important; in the latter, especially in applications to algebra, many of the techniques developed for proving these theorems have been useful.

In this paper I prove several preservation theorems for first order logic which I discovered while investigating the asymptotic growth of classes of finite combinatorial structures. The significance of these theorems lies in their applications to problems in finite combinatorics. Since the applications require combinatorial and analytical techniques that are not pertinent to logical questions discussed here, I shall present them in another paper [7].