It is a well-known fact that two structures are ∞ω-equivalent if and only if they are isomorphic in some Boolean extension of the universe of sets (cf. [4]; an early allusion to this result appears in [8]). My principal object here is to show that arbitrary toposes defined over the category of sets may be used instead. Thus ∞ω-equivalence means isomorphism in the extremely general context of some universe of "variable" sets in which not only is much of the usual set-theoretic machinery unavailable but the underlying logic is not even classical. This provides further support for the view that ∞ω-equivalence is a relation between structures of fundamental importance.

In this paper we show how some of the finite injury priority arguments can be simplified by making explicit use of the primitive notions of axiomatic computational complexity theory. Phrases such as “perform n steps in the enumeration of Wi” certainly bear witness to the fact that many of these complexity notions have been used implicitly from the early days of recursive function theory. However, other complexity notions such as that of an “honest” function are not so apparent, neither explicitly nor implicitly. Accordingly, one of the main factors in our simplification of these diagonalization arguments is the replacement of the characteristic function χA of a set A by the function νA, which is the next-element function of the set A. Another important factor is the use of busy beaver sets (see [3]) to provide the basis for the required diagonalizations thereby permitting rather simple and explicit descriptions of the sets constructed. Although the differences between the priority method and our method of construction are subtle, they are nonetheless real and noteworthy.

In preparation for the results which follow we devote the remainder of this section to the requisite definitions and notions as well as some preliminary lemmas. A more comprehensive discussion of many of the notions in this section can be found in [3]. Since we will be dealing extensively with relative computations most of our notions here have been correspondingly relativized.

We exhibit a finite list of first-order axioms which may be used to define topological spaces. For most separation axioms we discover a first-order equivalent statement.

We discuss the problem of defining the collection of first-order elementary classes in terms of the natural topological space of countable models.

This paper is a contribution to the growing literature on the model theory of nilpotent groups. (See Baumslag and Levin [2]; Eršov [5]; Hodges [9], [10]; Mal′cev [14]; Olin [16] and Saracino [19], [20].) In it we investigate the conditions under which the free product in the variety of all nilpotent of class 2 (nil-2) groups preserves saturation and stability.

It is well known that the direct product preserves both saturation (see Waszkiewicz and Wȩglorz [23]) and stability (see Wierzejewski [24]; Macintyre [13]; Eklof and Fisher [4]). On the other hand it is easy to show that the full free product of groups preserves neither property; indeed, in the case of saturation this failure is extremely bad since no free product of nontrivial groups is even 2-saturated. Our results show that the nil-2 free product falls between these two extremes.

Our proofs are mainly model-theoretic with a smattering of elementary algebra and rely heavily upon the unique normal form for the elements of a nil-2 free product given by MacHenry in [12]. (This normal form and some of its consequences are discussed in §1.) We assume familiarity with the basic ideas of saturation (see Chapter 5 of [3]) and Shelah's treatment of stability in [22].

We prove two main theorems in §3 each giving a necessary and sufficient condition in separate situations for the preservation of saturation. In the first (Theorem 3.1) we allow one finite factor, while in the second (Theorem 3.10) we deal solely with torsion groups. Our motivation for the proof of sufficiency was the paper of Waszkiewicz and Wȩglorz [23] and the principal tool is a “Feferman-Vaught” Theorem for the nil-2 free product which we prove in §2. We also show that if both factors in a nil-2 free product are nontorsion and one factor has a nil-2 basis, then the group is not even 3-saturated. We leave open the case where both factors are infinite but only one is torsion.

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.

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.

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.

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.

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.

A Boolean algebra (henceforth abbreviated B.A.) is said to be recursive if B is a recursive subset of the natural numbers N and the operations ∧ (meet), ∨ (join), and ¬ (complement) are partial recursive. Let denote the set of atoms of and denote the ideal generated by the atoms of . Given recursive B.A.s and , we write ≈ if is isomorphic to and ≈r if is recursively isomorphic to , i.e., if there is a partial recursive isomorphism from onto .

Recursive B.A.s have been studied by several authors including Ershov [2], Fiener [3], [4], Goncharov [5], [6], [7], LaRoche [8], Nurtazin [7], and the author [10], [11]. This paper continues a study of the recursion theoretic relationships among , , and the recursive isomorphism type of a recursive B.A. we started in [11]. We refer the reader to [11] for any unexplained notation and definitions. In [11], we were mainly concerned with the possible recursion theoretic properties of the set of atoms in recursive B.A.s. We found that even if we insist that be recursive, there is considerable freedom for the properties of . For example, we showed that if is a recursive B.A. such that is recursive and is infinite, then (i) there exists a recursive B.A. such that and both and are recursive and (ii) for any nonzero r.e. degree δ, there exist recursive B.A.s , , … such that for each i, is of degree δ, is recursive, is immune if i is even and is not immune if i is odd, and no two B.A.s in the sequence are recursively isomorphic.

The study of effectiveness in classical mathematics is rapidly expanding, through recent research in algebra, topology, model theory, and functional analysis. Well-known contributors are Barwise (Wisconsin), Crossley (Monash), Dekker (Rutgers), Ershoff (Novosibirsk), Feferman (Stanford), Harrington (Berkeley), Mal′cev (Novosibirsk), Morley (Cornell), Nerode (Cornell), Rabin (Hebrew University), Shore (Cornell). Further interesting work is due to Kalantari (University of California, Santa Barbara), Metakides (Rochester), Millar (Wisconsin), Remmel (University of California, San Diego), Nurtazin (Novosibirsk). Areas investigated include enumerated algebras, models of complete theories, vector spaces, fields, orderings, Hilbert spaces, and boolean algebras.

We investigate the effective content of the structure theory of p-groups. Recall that a p-group is a torsion abelian group in which the (finite) order of each element is some power of a fixed prime p. (In the sequel, “group” = “additively written abelian group”.)

The structure theory of p-groups is based on the two elementary notions of order and height. Recall that the order of x is the least integer n such that nx = 0. The height of x is the number of times p divides x, that is, the least n such that x = pny for some y in the group but x ≠ pn+1y for any y. If for each n ∈ ω there is a “pnth-root” yn, so that x = pnyn, then we say that x has infinite height. In 1923, Prüfer related the two notions as criteria for direct sum decomposition, proving

Theorem. Every group of bounded order is a direct sum of cyclic groups, and

Theorem. Every countable primary group with no (nonzero) elements of infinite height is a direct sum of cyclic groups.

We define a first-order theory FIN which has a recursive axiomatization and has the following two properties. Each finite part of FIN has finite models. FIN is strong enough to develop that part of mathematics which is used or has potential applications in natural science. This work can also be regarded as a consistency proof of this hitherto informal part of mathematics. In FIN one can count every set; this permits one to prove some new probabilistic theorems.

In this paper we are concerned with the formal provability of certain true implications of Σ1-sentences. Old completeness and incompleteness results already give some information about this. For example by Σ1-completeness of PRA (primitive recursive arithmetic) every true implication of the form D → E, where D is a Σ0-sentence and E a Σ1-sentence, or D a Σ1-sentence and E a true Σ1-sentence, is provable in PRA. On the other hand, by Gödel's incompleteness theorems one can define for every suitable theory S a false Σ1-sentence Ds such that for every false Σ0-sentence E the true implication Ds → E is not provable in S, but is provable in PRA + Cons. So one sees that for a suitable fixed conclusion the provability of true implications of Σ1-sentences depends on the content of the premise.

Now we ask, if and how for a suitable fixed premise the provability of true implications of Σ1-sentences depends on the conclusion. As remarked above, by Σ1-completeness of PRA this question is settled, if the premise is true. For a false premise it is answered in § 1 as follows: Let D be a false Σ1-sentence, S an extension of PRA, and S+ ≔ PRA + IAΣ1 + RFNΣ1S). Depending on D and S one can define a Σ1-sentence Es such that S+ ⊢ D → Es, but S ⊬ D → Es provided that S+ is not strong enough to refute D. (IAΣ1 denotes the scheme of the induction axiom for Σ1-formulas, and RFNΣ1(S) the uniform Σ1-reflection principle for S.)

In this paper we prove, under the assumption of a strong partition property for an uncountable cardinal κ, the existence of more than κ-many measurable cardinals greater than κ. Our proof involves so-called seminormal measures, and, along the way, we establish several key facts about such measures.

Quantification theory, or first-order predicate logic, can be formulated in terms purely of predicate letters and a few predicate functors which attach to predicates to form further predicates. Apart from the predicate letters, which are schematic, there are no variables. On this score the plan is reminiscent of the combinatory logic of Schönfinkel and Curry. Theirs, however, had the whole of higher set theory as its domain; the present scheme stays within the bounds of predicate logic.

In 1960 I published an apparatus to this effect, and an improved version in 1971. In both versions I assumed two inversion functors, major and minor; also a cropping functor and the obvious complement functor. The effects of these functors, when applied to an n-place predicate, are as follows:

The variables here are explanatory only and no part of the final notation. Ultimately the predicate letters need exponents showing the number of places, but I omit them in these pages.

A further functor-to continue now with the 1971 version-was padding:

Finally there was a zero-place predicate functor, which is to say simply a constant predicate, namely the predicate ‘I’ of identity, and there was a two-place predicate functor ‘∩’ of intersection. The intersection ‘F ∩ G’ received a generalized interpretation, allowing ‘F’ and ‘G’ to be predicates with unlike numbers of places. However, Steven T. Kuhn has lately shown me that the generalization is unnecessary and reducible to the homogeneous case.

The text of this article was done together with Gödel in 1976 to 1977 and was approved by him at that time. The footnotes and section headings have been added much later.

Gödel was born on April 28, 1906 at Brno (or Brünn in German), Czechoslovakia (at that time part of the Austro-Hungarian Monarchy). After completing secondary school there, he went, in 1924, to Vienna to study physics at the University. His interest in precision led him from physics to mathematics and to mathematical logic. He enjoyed much the lectures by Furtwangler on number theory and developed an interest in this subject which was, for example, relevant to his application of the Chinese remainder theorem in expressing primitive recursive functions in terms of addition and multiplication. In 1926 he transferred to mathematics and coincidentally became a member of the M. Schlick circle. However, he has never been a positivist, but accepted only some of their theses even at that time. Later on, he moved further and further away from them. He completed his formal studies at the University before the summer of 1929. He also attended during this period philosophical lectures by Heinrich Gomperz whose father was famous in Greek philosophy.

At about this time he read the first edition of Hilbert-Ackermann (1928) in which the completeness of the (restricted) predicate calculus was formulated and posed as an open problem. Gödel settled this problem and wrote up the result as his doctoral dissertation which was finished and approved in the autumn of 1929. The degree was granted on February 6, 1930. A somewhat revised version of the dissertation was published in 1930 in the Monatshefte.