On sait, depuis les travaux de Zil'ber et de Cherlin, que le degré de Morley de la théorie T d'un groupe G totalement transcendant est l'indice du plus petit sous-groupe définissable d'indice fini de G. Il est clair qu'il lui est supérieur, et l'inégalité inverse peut s'obtenir de la manière suivante: on fait agir G sur les types de S1(G) de rang de Morley maximum en associant à p, type de x au-dessus de G, le type ap de ax au-dessus de G; on montre alors que cette action est définissable, que le fait que ap = q équivaut au fait que a satisfasse une certaine formule à paramètres dans G, ce qui est bien facile si on n'oublie pas que dans une théorie stable tous les types sont définissables; on montre ensuite que cette action est transitive, que si p et q sont de rang de Morley maximum il existe a dans G tel que ap = q, et la méthode la plus rapide, mais qui est aussi la plus sophistiquée, est d'utiliser l'argument de symétrie de la déviation employé dans la preuve de la Proposition 1 de présent article; on conclut alors puisque le degré de Morley, qui est par définition le nombre de types de rang de Morley maximum, ést egal à l'indice du stabilisateur de p, qui est définissable.

Ce comportement des types de rang de Morely maximum se retrouve sans peine, si G est seulement superstable, dans celui des types de “plus petit rang continu” (encore appellé “degré de Shelah”) maximum. Pour trouver ce qui leur correspond dans le cas où G est seulement stable, il faut être un peu plus soigneux, et considérer les types p de S1(G), où G aura éventuellement été remplacé par une extension élémentaire suffisament saturée, tels que pour tout a de G ap ne dévie pas sur ∅: on montre qu'ils existent, et qu'ils sont tous conjugués par action de G; le fait que ap = q s'exprimera cette fois par une infinité de formules et non plus par une seule.

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.

This paper is concerned with algebraic independence in structures that are relatively simple for their size. It is shown that for κ a limit cardinal, if a structure of power at least κ is ∞ω-equivalent to a structure of power less than κ, then must contain an infinite set of algebraically independent elements. The same method of proof yields the fact that if σ is an Lω1ω-sentence (not necessarily complete) and σ has a model of power ℵω then some model of σ contains an infinite algebraically independent set.

All structures are assumed to be of countable similarity type. Letters , etc. will be used to denote either a structure or the universe of the structure. If X ⊆ , the algebraic closure of X (in ), denoted by Cl(X), is the union of all finite sets that are weakly definable (in ) by Lωω-formulas with parameters from X. A set S is algebraically independent if for each a in S, a ∉ Cl(S – {a}). An algebraically independent set is sometimes called a “free” set (in [3] and [4], for example).

It is known (see [5]) that any structure of power ℵn must have a set of n algebraically independent elements, and there are structures of power ℵn with no independent set of size n + 1. In power ℵω every structure will have arbitrarily large finite algebraically independent sets. However, it is consistent with ZFC that some models of power ℵω do not have any infinite algebraically independent set. Devlin [4] showed that if V = L, then for any cardinal κ, if every structure of power κ has an infinite algebraically independent set, then κ has a certain large cardinal property that ℵω can never possess.

A 2-affable ultrafilter has only finitely many predecessors in the Rudin-Keisler ordering of isomorphism classes of ultrafilters over the natural numbers. If the continuum hypothesis is true, then there is an ℵ1-sequence of ultrafilters Dα such that the strict Rudin-Keisler predecessors of Dα are precisely the isomorphs of the Dβ's for β < α.

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.

Krull [4] extended Galois theory to arbitrary normal extensions, in which the Galois groups are precisely the profinite groups (i.e. totally disconnected, compact, Hausdorff groups). Metakides and Nerode [7] produced two recursively presented algebraic extensions K ⊂ F of the rationals such that F is abelian, F is of infinite degree over K, and the Galois group of F over K, although of cardinality c, has only one recursive element (viz. the identity). This indicated the limits of effectiveness for Krull's theory. (The Galois theory of finite extensions is completely effective.) Nerode suggested developing a natural effective version of Krull's theory (done here in §1).

It is evident from the classical literature that the free profinite group on denumerably many generators can be obtained effectively as the Galois group of a recursive extension of the rationals over a subfield. Nerode conjectured that it could be obtained effectively as the Galois group of the algebraic numbers over a suitable subfield (done here in §2). The case of finitely many generators was done non-effectively by Jarden [3]. The author believes that the denumerable case, as presented in §2, is also new classically. Using this result and the effective Krull theory, every “co-recursively enumerable” profinite group is effectively the Galois group of a recursively enumerable field of algebraic numbers over a recursive subfield.

Assume (∃κ) [κ → (κ)<ω & Rκ ≺ V]. Then a new inner model H exists and has the following properties: (1) H ≠ HOD; (2) Th(H) = Th(HOD); (3) there is j: H → H; (4) there is a c.u.b. class of indiscernibles for H.

In this paper we investigate the relationship between the number of countable and decidable models of a complete theory. The number of decidable models will be determined in two ways, in §1 with respect to abstract isomorphism type, and in §2 with respect to recursive isomorphism type.

Definition 1. A complete theory is (α, β) if the number of countable models of T, up to abstract isomorphism, is β, and similarly the number of decidable models of T is α.

Definition 2. A model is ω-decidable if ∣∣= ω and for an effective listing {θn∣n < ω} of all sentences in the language of Th() augmented by new constant symbols i*, i < ω, {n ∣〈, i〉i<ω ⊨ θn} is recursive, where i interprets i* (in these terms, is decidable if is abstractly isomorphic to an ω-decidable model).

Definition 3. A complete theory is (α, β)r if it is (γ, β) for some γ and it has exactly αω-decidable models up to recursive isomorphism.

Specifically we will show in §1 that there is a (2, ω) theory, and in §2 that although there is a (2, 2ω) theory, there is no (2, β)r theory for any β, β < 2ω.