Though researchers in the field of abstract computational complexity theory have utilized many of the tools of recursive function theory in the development of their field, the early results obtained (e.g., see [8]) seemed to be rather independent of results in recursion theory (at least to the extent that the results were not uniformly interesting to both varieties of theorists). It seems to have been generally accepted, however, that strong parallels of one form or another must exist between the two fields. Indeed, recent results of Blum and Marques [7], Morris [13], Soare [15] and Bennison [1], [3] have revealed a striking correspondence between complexity theoretic properties and recursion theoretic properties. These results are not contrived, but rather link together interesting properties which had arisen naturally and independently in their respective fields. This paper presents the results of research aimed at finding a recursion theoretic characterization for a complexity theoretic property which had arisen from the study of the speed-up phenomenon.

In abstract computational complexity theory we are concerned with categorizing computable functions or sets according to their relative difficulty of computation. The phrase “difficult to compute” may take on different meanings depending on which criteria (complexity theoretic properties) we use to define what it means for a function or set to be hard to compute. In the abstract setting, however, such criteria should yield the same classes of functions or sets regardless of the underlying abstract complexity measure (in the sense of Blum [4], e.g., tape, time, etc.). In other words, such criteria should be measure-independent. In this paper we will be considering one way of defining “difficult to compute”. Namely, we shall say that a function or set is difficult to compute if it does not have a recursively enumerable complexity sequence as defined by Meyer and Fischer [12]. For a property to have a recursion theoretic characterization it must be measure-independent, for a recursion theoretic property is, by its very nature, measure-independent. It will not be immediately obvious whether or not the property of having an r.e. complexity sequence is measure-independent. We attack this question by first considering an alternative definition of an r.e. complexity sequence, one which is easily seen to be measure-independent.

Building on Hrushovski's work in [5], we study definable groupoids in stable theories and their relationship with 3-uniqueness and finite internal covers. We introduce the notion of retractability of a definable groupoid (which is slightly stronger than Hrushovski's notion of eliminability), give some criteria for when groupoids are retractable, and show how retractability relates to both 3-uniqueness and the splitness of finite internal covers. One application we give is a new direct method of constructing non-eliminable groupoids from witnesses to the failure of 3-uniqueness. Another application is a proof that any finite internal cover of a stable theory with a centerless liaison groupoid is almost split.

We construct a computable ℵ0-categorical structure whose first order theory is computably equivalent to the true first order theory of arithmetic.

We show that in the lattice of classes there are initial segments [∅, P] = (P) which are not Boolean algebras, but which have a decidable theory. In fact, we will construct for any finite distributive lattice L which satisfies the dual of the usual reduction property a class P such that L is isomorphic to the lattice (P)*, which is (P). modulo finite differences. For the 2-element lattice, we obtain a minimal class, first constructed by Cenzer, Downey, Jockusch and Shore in 1993. For the simplest new class P constructed, P has a single, non-computable limit point and (P)* has three elements, corresponding to ∅, P and a minimal class P0 ⊂ P, The element corresponding to P0 has no complement in the lattice. On the other hand, the theory of (P) is shown to be decidable.

A class P is said to be decidable if it is the set of paths through a computable tree with no dead ends. We show that if P is decidable and has only finitely many limit points, then (P)* is always a Boolean algebra. We show that if P is a decidable class and (P) is not a Boolean algebra, then the theory of (P) interprets the theory of arithmetic and is therefore undecidable.

One of the main goals in the theory of forcing iteration is to formulate preservation theorems for not collapsing ω1 which are as general as possible. This line leads from c.c.c. forcings using finite support iterations to Axiom A forcings and proper forcings using countable support iterations to semi-proper forcings using revised countable support iterations, and more recently, in work of Shelah, to yet more general classes of posets. In this paper we concentrate on a special case of the very general iteration theorem of Shelah from [5, chapter XV]. The class of posets handled by this theorem includes all semi-proper posets and also includes, among others, Namba forcing.

In [5, chapter XV] Shelah shows that, roughly, revised countable support forcing iterations in which the constituent posets are either semi-proper or Namba forcing or P[W] (the forcing for collapsing a stationary co-stationary subset of

with countable conditions) do not collapse ℵ1. The iteration must contain sufficiently many cardinal collapses, for example, Levy collapses. The most easily quotable combinatorial application is the consistency (relative to a Mahlo cardinal) of ZFC + CH fails + whenever A ∪ B = ω2 then one of A or B contains an uncountable sequentially closed subset. The iteration Shelah uses to construct this model is built using P[W] to “attack” potential counterexamples, Levy collapses to ensure that the cardinals collapsed by the various P[W]'s are sufficiently well separated, and Cohen forcings to ensure the failure of CH in the final model.

In this paper we give details of the iteration theorem, but we do not address the combinatorial applications such as the one quoted above.

These theorems from [5, chapter XV] are closely related to earlier work of Shelah [5, chapter XI], which dealt with iterated Namba and P[W] without allowing arbitrary semi-proper forcings to be included in the iteration. By allowing the inclusion of semi-proper forcings, [5, chapter XV] generalizes the conjunction of [5, Theorem XI.3.6] with [5, Conclusion XI.6.7].

The notions of forcing and generic set were introduced by Cohen in 1963 to prove the independence of the Axiom of Choice and the Continuum Hypothesis in set theory. Let ω be the set of natural numbers, i.e., {0,1,2,3,…}. A string is a mapping from an initial segment of ω into {0,1}. We identify a set A ⊆ ω to with its characteristic function.

We now consider a set generic over the arithmetic sets. A set A ⊆ ω is called n-generic if it is Cohen-generic for n-quantifier arithmetic. This is equivalent to saying that for every -set of strings S, there is a σ ⊂ A such that σ ∈ S or (∀v ≥ σ)(v ∉ S). By degree we mean Turing degree (of unsolvability). We call a degree n-generic if it has an n-generic representative. For a degree a, let D(≤a) denote the set of degrees which are recursive in a.

Before Cohen's work, there was a precursor of the notion of forcing in recursion theory. Friedberg showed that for every degree b above the complete degree 0', i.e., the degree of a complete r.e. set, there is a degree a such that a′ = a ⋃ 0′ = b. He actually proved this result by using the notion of forcing for statements.

In their paper [3], Hajnal and Komjáth define the following combinatorial principle:

Definition 1.1. Suppose κ is an infinite cardinal and n < ω. Then Hn(κ) is the statement: There is a function F: [κ]n → [[κ]ω]≤ω such that

(a) ∀A ∈[κ]n ∀Y ∈ F(A)(Y ⊆ min (A)), and

(b) .

Hn(κ) is related to a more general principle introduced by Hajnal and Nagy in [4]. For applications of these principles to free sets for set mappings and Ramsey games we refer the reader to [3] and [4].

In [3] Hajnal and Komjáth prove the consistency of ZFC + GCH + ∀n ∈ ω(Hn + 1(ωn + 1)), relative to an ω-Mahlo cardinal. They conjecture that L is a model of this theory, and suggest that the proof might require higher gap morasses. The first few cases of this conjecture are known to be true; it is easy to see that if CH holds then H1 (ω1) is true, and Laver proved that V = L implies H2(ω2). In this paper we go one step further and prove V = L → H3(ω3). Unfortunately our methods do not appear to give Hn (ωn) for n ≥ 4.

Most of our notation is standard. If X is any set and κ is a cardinal number then [X]κ is the set of subsets of X with cardinality κ, and [X]≤κ is the set of subsets of X with cardinality ≤ κ. If X is a set of ordinals then tp(X) is the order type of X.

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.

We establish several first- or second-order properties of models of first-order theories by considering their elements as atoms of a new universe of set theory and by extending naturally any structure of Boolean model on the atoms to the whole universe. For example, complete f-rings are “boundedly algebraically compact” in the language (+, −, ·, ∧, ∨, ≤), and the positive cone of a complete l-group with infinity adjoined is algebraically compact in the language (+, ∨, ≤). We also give an example with any first-order language. The proofs can be translated into “naive set theory” in a uniform way.

Let PA be Peano arithmetic with function symbols′, + and ·. The length of a proof P, denoted by lh(P), is the maximum length of threads of P (for the term ‘thread’, see [T, p. 14]). For a formula A and a natural number m, PA ⊢mA denotes the fact that there is a proof in PA of A whose length is less than or equal to m. PA ⊢ A denotes the fact that there is a proof in PA of A.

G. Kreisel conjectured that the following proposition holds.

“Let m be a natural number and A(a) be a formula. If for each natural number n, then PA ⊢ ∀xA(x)”.

Let PA1 be the corresponding system with + and · replaced by ternary predicates and axioms saying that these predicates represent functions. Parikh [P] proved the following proposition which is obtained from Kreisel's conjecture by replacing PA by PA1.

Proposition. Let A(a) be a formula in PA1and m be a natural number. Assume thatfor each natural number n. Then PA1 ⊢ ∀xA(x).

The reason why Parikh's method succeeds is the fact that the only function symbol ′ in PA1 is unary. So his method fails for PA.

To solve this conjecture for PA, we must make syntactical investigation into proofs in PA of formulas of the form A() with length ≤ m. Even if lengths of proofs are less than or equal to m, depths of occurrences of bound variables in induction axiom schemata or equality axiom schemata in proofs are not always bounded.

This is the first part of a series of papers on certain topics of model theory of modules, which will further consist of Part II. On stability and categoricity of flat modules and Part III. On infiniteness of sets definable in modules. Although these parts are only weakly connected, it is convenient to have common background and notation. The first part was written in Summer 1980, when I did not know about either footnote (3) appearing in the final version of [GA 2] or about the work of the “Bedford College Group” concerning stability theory of modules. I would like to thank Wilfrid Hodges and Mike Prest for their interest in this note, which encouraged me to include it as the first part of the present series, and for acquainting me with their unpublished work, to which I refer in footnotes. I should also like to thank the referee for his suggestions, in particular for correcting the argument in Remark (5) and for drawing my attention to [Z-H].

A syntactical analysis of Zimmermann's notion of “endlich matriziellen Untergruppen” of modules [ZI, p. 1087] shows that it coincides with the notion of subgroups definable by positive primitive formulae (without parameters) [GA 1, p. 80]. Then combination of [ZI, Folgerung 3.4] with [GA 1, Lemma 5] (or [GA 2, Theorem 1] for uncountable rings) yields purely algebraic criteria for total transcendence of modules. I present here a proof of this result using only the following facts.

(G1) A module M is totally transcendental iff it satisfies the minimal condition on subgroups definable by positive primitive formulae (without parameters) [GA 1, Lemma 5] (and [GA 2, Theorem 1] for uncountable rings).

We generalize the ultrapower in a way suitable for choiceless set theory. Given an ultrafilter, forcing is used to construct an extended ultrapower of the universe, designed so that the fundamental theorem of ultrapowers holds even in the absence of the axiom of choice. If, in addition, we assume DC, then an extended ultrapower of the universe by a countably complete ultrafilter must be well-founded. As an application, we prove the Vopěnka-Hrbáček theorem from ZF + DC only (the proof of Vopěnka and Hrbáček used the full axiom of choice): if there exists a strongly compact cardinal, then the universe is not constructible from a set. The same method shows that, in L[2ω], there cannot exist a θ-compact cardinal less than θ (where θ is the least cardinal onto which the continuum cannot be mapped); a similar result can be proven for other models of the form L[A]. The result for L[2ω] is of particular interest in connection with the axiom of determinacy. The extended ultrapower construction of this paper is an improved version of the author's earlier pseudo-ultrapower method, making use of forcing rather than the omitting types theorem.

Let Kr be the class of all those quantificational formulas whose matrices are conjunctions of binary disjunctions of signed atomic formulas. Decision problems for subclasses of Kr do not invariably coincide with those for the corresponding classes of quantificational formulas with unrestricted matrices; for example, the ∀∃∀ prefix subclass of Kr is solvable, but the full ∀∃∀ class is not ([AaLe],- [KMW]). Moreover, the straightforward encodings of automata which have been used to show the unsolvability of various subclasses of Kr ([Aa], [Bö], [AaLe]) yield but little information about signature subclasses, i.e. subclasses determined by the number and degrees of the predicate letters occurring in a formula. By a new and highly complex construction Theorem 1 establishes the best possible result on classification by signature.

Theorem 1. Let C be the class of all formulas in Kr with a single predicate letter, which is dyadic; then C is a reduction class for satisfiability.

Thus a signature subclass of Kr is solvable just in case the corresponding class of unrestricted quantificational formulas is solvable, to wit, just in case no predicate letter of degree exceeding one may occur. To obtain a richer classification by signature we consider further restrictions on the matrix. Let Or be the class of formulas in Kr having disjunctive normal forms with only two disjuncts. Theorem 2 sharpens Orevkov's proof of the unsolvability of Or ([Or]; see also [LeGo]).

Theorem 2. Let D be the class of formulas in Or with just two predicate letters, both pentadic; then D is a reduction class for satisfiability.

Given a family ${\cal C}$ of infinite subsets of ${\Bbb N}$, we study when there is a Borel function $S:2^{\Bbb N} \to 2^{\Bbb N} $ such that for every infinite $x \in 2^{\Bbb N} $, $S\left( x \right) \in {\Cal C}$ and $S\left( x \right) \subseteq x$. We show that the family of homogeneous sets (with respect to a partition of a front) as given by the Nash-Williams’ theorem admits such a Borel selector. However, we also show that the analogous result for Galvin’s lemma is not true by proving that there is an $F_\sigma $ tall ideal on ${\Bbb N}$ without a Borel selector. The proof is not constructive since it is based on complexity considerations. We construct a ${\bf{\Pi }}_2^1 $ tall ideal on ${\Bbb N}$ without a tall closed subset.

We define a rich model to be one which contains a proper elementary substructure isomorphic to itself. Existence, nonstructure, and categoricity theorems for rich models are proved. A theory T which has fewer than min(2λ, ℶ2) rich models of cardinality λ (λ > ∣T∣) is totally transcendental. We show that a countable theory with a unique rich model in some uncountable cardinal is categorical in ℵ1 and also has a unique countable rich model. We also consider a stronger notion of richness, and use it to characterize superstable theories.

The complete diagram of a structure , denoted by Dc(), is the set of all sentences true in the structure (, a)a∈. A structure is said to be resplendent if for every sentence θ involving a new relation symbol R in addition to symbols occurring in Dc(), if θ is consistent with Dc(), then there is a relation P on such that (see[1]).

Baldwin asked whether a homogeneous recursively saturated structure is necessarily resplendent. Here it is shown that this need not be the case. It is shown that if is an uncountable homogeneous resplendent model of an unstable theory, then must be saturated. The proof is related to the proof in [5] that an uncountable homogeneous recursively saturated model of first order Peano arithmetic must be saturated. The example for Baldwin's question is an uncountable homogeneous model for a particular unstable theory, such that is recursively saturated and omits some type. (The continuum hypothesis is needed to show the existence of such a model in power ℵ1.)

The proof of the main result requires two lemmas.

We generalise the α-Ramsey cardinals introduced in Holy and Schlicht (2018) for cardinals α to arbitrary ordinals α, and answer several questions posed in that paper. In particular, we show that α-Ramseys are downwards absolute to the core model K for all α of uncountable cofinality, that strategic ω-Ramsey cardinals are equiconsistent with remarkable cardinals and that strategic α-Ramsey cardinals are equiconsistent with measurable cardinals for all α > ω. We also show that the n-Ramseys satisfy indescribability properties and use them to provide a game-theoretic characterisation of completely ineffable cardinals, as well as establishing further connections between the α-Ramsey cardinals and the Ramsey-like cardinals introduced in Gitman (2011), Feng (1990), and Sharpe and Welch (2011).