Silver and subsequently Galvin and Hajnal, got bounds on , for ℵα strong limit cardinal of cofinality > ℵ0. We somewhat improve those results.

Sanchis [Sa] and Diller [Di] have introduced an interesting relation between terms for primitive recursive functionals of finite type in order to obtain a computability proof. The method is as follows. First the relation is shown to be well-founded. Then computability of each term is obtained by transfinite induction over the relation.

Their relation is given by various clauses which define the (immediate) successors of a term. Hence, starting with a term H and repeatedly applying the successor relation one generates the tree of H (say). The problem is to prove that the trees are well-founded. In their proofs Sanchis and Diller use the axiom of bar induction. Diller also gives a proof using transfinite induction: each tree is shown to have a length b < φω(0) (the first ω-critical number) and the induction is over ordinals less than or equal to b.

For proof-theoretic purposes it would be desirable to employ a method of proof which is more elementary than the axiom of bar induction and also to obtain a smaller ordinal bound. The purpose of the following is to provide such a method. We show that for a primitive recursive term H of finite type the tree generated by the relation of Sanchis and Diller has length less than ε0. Our method is similar to that used by Tait [Ta] in his theory of infinite terms. The immediately natural metamathematics is elementary intuitionistic analysis with the axiom of choice and transfinite induction over the relevant ordinal notations. We give several versions of the basic list of successor clauses, and for one version we show how to carry out the treatment in Skolem arithmetic (with primitive recursion of lowest type).

Theorem 1. (◊ℵ1,) If B is an infinite Boolean algebra (BA), then there is B1, such that ∣ Aut (B1) ≤∣B1∣ = ℵ1 and 〈B1, Aut (B1)〉 ≡ 〈B, Aut(B)〉.

Theorem 2. (◊ℵ1) There is a countably compact logic stronger than first-order logic even on finite models.

This partially answers a question of H. Friedman. These theorems appear in §§1 and 2.

Theorem 3. (a) (◊ℵ1) If B is an atomic ℵ-saturated infinite BA, Ψ Є Lω1ω and 〈B, Aut (B)〉 ⊨Ψ then there is B1, Such that ∣Aut(B1)∣ ≤ ∣B1∣ =ℵ1, and 〈B1, Aut(B1)〉⊨Ψ. In particular if B is 1-homogeneous so is B1. (b) (a) holds for B = P(ω) even if we assume only CH.

We prove the independence of a strong partition relation on ℵω, answering a question of Erdös and Hajnal. We then give an almost complete answer to the free subset problem.

In this paper we prove that Beth's property does not hold in L<ω. This answers a question posed by Magidor and Malitz in [8]. Beth's property is a natural closure condition on a language which says that everything implicitly definable in the language is also explicitly definable in the language. That the first-order predicate calculus (L0) has the property was first shown by Beth [4]. Lopez-Escobar proved that also has Beth's property [7]. Malitz [9] showed the Beth's property fails in Lκλ where κ ≥ λ ≥ ω1. Friedman and Silver showed that the property fails in Lκλ for κ > ω1. Also Friedman [5] showed that extensions of elementary logic using cardinality quantifiers (L)1κ do not have Beth's property.

The undefined notation used here is standard. If further clarification is needed, we refer the reader to [3]. κ and λ denote infinite cardinals. cX and ȣX∣ denote the cardinality of X. cfμ will denote the cofinality of the order μ. All languages discussed are assumed to have no function or constant symbols. All structures are relational. The type of a formula is the set of relation symbols appearing in the formula (excluding equality). The type of a set of formulas is the set of all relation symbols appearing in some formula of the set of formulas. For purpose of this paper we will assume that to each n-ary relation in a structure there is associated an n-ary relation symbol and that in any given structure this association is one-to-one. Using this convention we can define the type of a structure to be the set of those relation symbols to which there is associated a relation in the structure. Sometimes we use the term predicate instead of relation.

The modal logic S5 has been formulated in Gentzen-style by several authors such as Ohnishi and Matsumoto [4], Kanger [2], Mints [3] and Sato [5]. The system by Ohnishi and Matsumoto is natural, but the cut-elimination theorem in it fails to hold. Kanger's system enjoys cut-elimination theorem, but, strictly speaking, it is not a Gentzen-type system since each formula in a sequent is indexed by a natural number. The system S5+ of Mints is also cut-free, and its cut-elimination theorem is proved constructively via the cut-elimination theorem of Gentzen's LK. However, one of his rules does not have the so-called subformula property, which is desirable from the proof-theoretic point of view. Our system in [5] also enjoys the cut-elimination theorem. However, it is also not a Gentzen-type system in the strict sense, since each sequent in this system consists of a pair of sequents in the usual sense.

In the present paper, we give a Gentzen-type system for S5 and prove the cut-elimination theorem in a constructive way. A decision procedure for S5 can be obtained as a by-product.

The author wishes to thank the referee for pointing out some errors in the first version of the paper as well as for his suggestions which improved the readability of the paper.

Modular machines were introduced in [1] and [2], where they were used to give simple proofs of various unsolvability results in group theory. Here we discuss the degrees of the halting, word, and confluence problems for modular machines, both for their own sake and in the hope that the results may be useful in group theory (see [4] for an application of a related result to group theory).

In the course of the analysis, I found it convenient to compare degrees of these problems for a Turing machine T and for a Turing machine T1 obtained from T by enlarging the alphabet but retaining the same quintuples (or quadruples). The results were surprising. The degree for a problem of T1 depends not just on the corresponding degree for T, but also on the degrees of the corresponding problems when T is restricted to a semi-infinite tape (both semi-infinite to the right and semi-infinite to the left). For the halting and confluence problems, the Turing degrees of the problems for these three machines can be any r.e. degrees. In particular the halting problem of T can be solvable, while that of T1 has any r.e. degree.

A machine M (in the general sense) consists of a countable set of configurations (together with a numbering, which we usually take for granted), a recursive subset of configurations called the terminal configurations, and a recursive function, written C ⇒ C′, on the set of configurations. If, for some n ≥ 0, we have C = C0 ⇒ C1 ⇒ … ⇒ Cn = C′, we write C → C′. We say M halts from C if C → C′ for some terminal C′.

Theorem. The Jacobson radical of an ℵ0-categorical associative ring is nilpotent.

Consider the following propositions:

(A) Every uncountable subset of contains an uncountable chain or antichain (with respect to ⊆).

(B) Every uncountable Boolean algebra contains an uncountable antichain (i.e., an uncountable set of pairwise incomparable elements).

Until quite recently, relatively little was known about these propositions. The oldest result, due to Kunen [4] and the author independently, asserts that if the Continuum Hypothesis (CH) holds, then (A) is false. In fact there is a counter-example 〈Aα: α < ω1〉 such that α < β implies Aβ −Aα is finite. Kunen also observed that Martin's Axiom (MA) + ¬CH implies that no such counterexample 〈Aα: α < ω1〉 exists.

Much later, Komjáth and the author [2] showed that ◊ implies the existence of several kinds of uncountable Boolean algebras with no uncountable chains or antichains. Similar results (but motivated quite differently) were obtained independently by Rubin [5]. Berney [3] showed that CH implies that (B) is false, but his algebra has uncountable chains. Finally, Shelah showed very recently that CH implies the existence of an uncountable Boolean algebra with no uncountable chains or antichains.

Except for Kunen's result cited above, the only result in the other direction was the theorem, due also to Kunen, that MA + ¬CH implies that any uncountable subset of with no uncountable antichains must have both ascending and decending infinite sequences under ⊆.

In a previous paper [9] a demonstrably consistent type-free system of combinatory logic CΔ was presented. This system contained a moderately strong exten-sionality principle, the usual equations of combinatory logic, Boolean propositional connectives, unrestricted quantifiers, an unrestricted abstraction principle (“comprehension axiom”), and a limited principle of excluded middle. It also contained elementary arithmetic in its entirety, avoiding arithmetical Gödel incompleteness [14] by being ω-complete. CΔ probably can be shown to contain certain important parts of mathematical analysis, such as the theory of continuous functions, which are contained by the somewhat similar (but nonextensional) system K′ [6], [7]. A stronger system CΓ, having these same properties and more, will be formulated in the present paper. Elsewhere [13] it will be shown that the system Q of my book, Elements of combinatory logic (Yale University Press, New Haven and London, 1974), referred to below as ECL, is essentially a subsystem of CΓ, so that the consistency of CΓ guarantees that of Q.

CΓ will be stronger than both CΔ and Q in having an operator ‘ɿ’ (inverted Greek iota) which denotes the inverse of equality in the sense that ‘ɿ(= a) = a’ is a theorem of the system. Thus ‘ɿ’ denotes a function or operator which operates on a unit class to give the only member of that class. (Here ‘=a’ denotes the unit class whose only member is denoted by ‘a’.) If uninverted Greek iota, ‘ι’, is used as an abbreviation for ‘=’, the above equation could be written as ‘ɿ(ιa) = a’. Here ‘ιa’ is analogous to Bertrand Russell's well-known notation ‘ιa’, denoting a unit class. This same operator ‘ɿ’ makes it possible to transform one-many or many-one relations into the corresponding functions or operators, a kind of transformation not usually available in systems of combinatory logic [1], [2]. It also provides a method for restricting functions by restricting the corresponding relations.

We use the spaces and n of complete types and of existential types to investigate various notions which appear in the theory of the algebraic structure of models.

We prove first that if T is a countable complete theory with n(T), the number of countable models of T, equal to three, then T is similar to the Ehrenfeucht example of such a theory. Woodrow [4] showed that if T is in the same language as the Ehrenfeucht example, T has elimination of quantifiers, and n(T) = 3 then T is very much like this example. All known examples of theories T with n(T) finite and greater than one are based on the Ehrenfeucht example. We feel that such theories are a pathological case. Our second theorem strengthens the main result of [2]. The theorem in the present paper says that if T is a countable theory which has a model in which all the elements of some infinite definable set are algebraic of uniformly bounded degree, then n(T) ≥ 4. It is known [3] that if n(T) > 1, then n(T) > 3, so our result is the first nontrivial step towards proving that n(T) ≥ ℵ0. We would also like, of course, to prove the result without the uniform bound on the finite degrees of the elements in the subset.

Theorem 2.1 is included in the author's Ph. D. thesis, as is a weaker version of Theorem 3.7. Thanks are due to Harry Simmons for his suggestions concerning the presentation of the material, and to Wilfrid Hodges for his advice while I was a Ph. D. student.

There is no algorithm for determining whether or not an equation is true in every 3-dimensional cylindric algebra. This theorem completes the solution to the problem of finding those values of α and β for which the equational theories of CAα and RCAβ are undecidable. (CAα and RCAβ are the classes of α-dimensional cylindric algebras and representable β-dimensional cylindric algebras. See [4] for definitions.) This problem was considered in [3]. It was known that RCA0 = CA0 and RCA1 = CA1 and that the equational theories of these classes are decidable. Tarski had shown that the equational theory of relation algebras is undecidable and, by utilizing connections between relation algebras and cylindric algebras, had also shown that the equational theories of CAα and RCAβ are undecidable whenever 4 ≤ α and 3 ≤ β. (Tarski's argument also applies to some varieties K ⊆ RCAβ with 3 ≤ β and to any variety K such that RCAα ⊆ K ⊆ CAα and 4 ≤ α.)

Thus the only cases left open in 1961 were CA2, RCA2 and CA3. Shortly there-after Henkin proved, in one of Tarski's seminars at Berkeley, that the equational theory of CA2 is decidable, and Scott proved that the set of valid sentences in a first-order language with only two variables is recursive [11]. (For a more model-theoretic proof of Scott's theorem see [9].) Scott's result is equivalent to the decidability of the equational theory of RCA2, so the only case left open was CA3.

Let A be an abelian group and B a pure injective pure extension of A. Then there is a homomorphic image C of B over A which is a pure injective hull of A; C can be constructed by using Zorn's lemma to find a suitable congruence on B. In a paper [4] which greatly generalises this and related facts about pure injectives, Walter Taylor asks (Problem 1.5) whether one can find a “construction” of C which is more concrete than the one mentioned above; he asks also whether the points of C can be explicitly described. In this note I return the answer No.

A natural way of studying the computability of an algebraic structure or process is to apply some of the theory of the recursive functions to the algebra under consideration through the manufacture of appropriate coordinate systems from the natural numbers. An algebraic structure A = (A; σ1,…, σk) is computable if it possesses a recursive coordinate system in the following precise sense: associated to A there is a pair (α, Ω) consisting of a recursive set of natural numbers Ω and a surjection α: Ω → A so that (i) the relation defined on Ω by n ≡α m iff α(n) = α(m) in A is recursive, and (ii) each of the operations of A may be effectively followed in Ω, that is, for each (say) r-ary operation σ on A there is an r argument recursive function on Ω which commutes the diagram

wherein αr is r-fold α × … × α.

This concept of a computable algebraic system is the independent technical idea of M.O.Rabin [18] and A.I.Mal'cev [14]. From these first papers one may learn of the strength and elegance of the general method of coordinatising; note-worthy for us is the fact that computability is a finiteness condition of algebra—an isomorphism invariant possessed of all finite algebraic systems—and that it serves to set upon an algebraic foundation the combinatorial idea that a system can be combinatorially presented and have effectively decidable term or word problem.

The continuous or countable functionals were independently denned by Kleene [9] and Kreisel [10]. They were intended as a suitable basis for constructive mathematics, and thus it is interesting to investigate various notions of recursion on the countable functionals.

There have been two main streams in this investigation, the study of countable recursion and the study of computability or Kleene-recursion.

Countable recursion is the theory of recursion on the associates. Gandy and Hyland [3] and Hyland [7] are good sources for the recent development of countable recursion.

This paper will mostly be concerned with Kleene-recursion on the countable functionals as denned in Kleene [8] and [9]. We assume some familiarity with the countable functionals and associates, as presented in Kleene [9], Bergstra [1] or any other paper on the subject.

Pioneering work with recursion in nonnormal objects was done by Grilliot [4], who proved that a functional F of type 2 is normal if and only if its 1-section (that is the set of functions recursive in F) is closed under ordinary jump, and if and only if F is continuous on 1-section (F).

Hinman [6] constructed a countable functional that is not recursively equivalent to a function, and thereby showed that recursion in nonnormal functionals is an extension of ordinary recursion in functions.

In [6], Hinman asked if there are functionals with topless 1-sections, i.e. with no maximal elements in the semi-lattice of degrees. This was answered in the affirmative by Bergstra [1], using a spoiling construction. Thus the class of 1-sections of functionals extends the class of 1-sections of functions.

In [6], G. Metakides and the author introduced a general model theoretic setting in which to study the lattice of r.e. substructures of a large class of recursively presented models . Examples included , the natural numbers with equality, 〈 Q, ≤ 〉, the rationals under the usual ordering, and a large class of n-dimensional partial orderings. In this setting, we were able to generalize many of the constructions of classical recursion theory so that the constructions yield the classical results when we specialize to the case of and new results when we specialize to other models. Constructions to generalize Myhill's Theorem on creative sets [8], Friedberg's Theorem on the existence of maximal sets [3], Dekker's Theorem on the degrees of hypersimple sets [2], and Martin's Theorem on the degrees of maximal sets [5] were produced in [6]. In this paper, we give constructions to generalize the Morley-Soare Splitting Theorem [7] and Lachlan's characterization of hyperhypersimple sets [4] in §2, constructions to generalize Lachlan's theorems on the existence of major subsets and r-maximal sets contained in maximal sets [4] in §3, and constructions to generalize Robinson's construction of r-maximal sets that are not contained in any maximal sets [11] and second-order maximal sets [12] in §4.

In §1 of this paper, we give the precise definitions of our model theoretic setting and deal with other preliminaries. Also in §1, we define the notions of “uniformly nonrecursive”, “uniformly maximal”, etc. which are the key notions involved in the generalizations of the various theorems that occur in §§2, 3 and 4.