# Showing papers in "Annals of Pure and Applied Logic in 1998"

••

TL;DR: Structures equipped with generic predicates and/or automorphisms are studied, and it is shown that in many cases they obtain simple theories.

Abstract: We study structures equipped with generic predicates and/or automorphisms, and show that in many cases we obtain simple theories. We also show that a bounded PAC field (possibly imperfect) is simple. 1998 Published by Elsevier Science B.V. All rights reserved.

158 citations

••

TL;DR: It is proved that every monadic second-order property of the unfolding of a transition system is a monadicsecond-orderproperty of the system itself.

Abstract: We prove that every monadic second-order property of the unfolding of a transition system is a monadic second-order property of the system itself. An unfolding is an instance of the general notion of graph covering. We consider two more instances of this notion. A similar result is possible for one of them but not for the other.

69 citations

••

TL;DR: This work considers Borel equivalence relations E induced by actions of the infinite symmetric group, or equivalently the isomorphism relation on classes of countable models of bounded Scott rank, and relates the descriptive complexity of the equivalence relation to the nature of its complete invariants.

Abstract: We consider Borel equivalence relations E induced by actions of the infinite symmetric group, or equivalently the isomorphism relation on classes of countable models of bounded Scott rank. We relate the descriptive complexity of the equivalence relation to the nature of its complete invariants. A typical theorem is that E is potentially Π^0_3 iff the invariants are countable sets of reals, it is potentially Π^0_4 iff the invariants are countable sets of countable sets of reals, and so on. The proofs use various techniques, including Vaught transforms, changing topologies, and the Scott analysis of countable models.

60 citations

••

TL;DR: It is proved that any computable partially ordered set is isomorphic to the spectrum of an intrinsically c.e. relation on a computable structure, and the isomorphism can be constructed in such a way that the image of the minimum element of the partially ordering set is computable.

Abstract: The spectrum of a relation R on a computable structure is the set of Turing degrees of the image of R under all isomorphisms between A and any other computable structure B . The relation R is intrinsically computably enumerable (c.e.) if its image under all such isomorphisms is c.e. We prove that any computable partially ordered set is isomorphic to the spectrum of an intrinsically c.e. relation on a computable structure. Moreover, the isomorphism can be constructed in such a way that the image of the minimum element (if it exists) of the partially ordered set is computable. This solves the spectrum problem. The theorem and modifications of its proof produce computably categorical structures whose expansions by finite number of constants are not computably categorical and, indeed, ones whose expansions can have any finite number of computable isomorphism types. They also provide examples of computably categorical structures that remain computably categorical under expansions by constants but have no Scott family.

39 citations

••

TL;DR: These proofs, together with an embedding of the type theory in a set theoretical system as carried out in Setter (1993), show that the proof theoretical strength of Martin-Lof's type theory with W-type and one universe is strong.

Abstract: We present well-ordering proofs for Martin-Lof's type theory with W-type and one universe. These proofs, together with an embedding of the type theory in a set theoretical system as carried out in Setter (1993) show that the proof theoretical strength of

39 citations

••

TL;DR: It is shown that a finite dimensional algebra is rich iff it is rectangularly dense and quasi-atomic; moreover, each of these conditions is also equivalent to a very natural condition of point density, and every finite dimensional (or locally finite dimensional) rich algebra of logic is representable.

Abstract: Henkin and Tarski proved that an atomic cylindric algebra in which every atom is a rectangle must be representable (as a cylindric set algebra). This theorem and its analogues for quasi-polyadic algebras with and without equality are formulated in Henkin, Monk and Tarski [13].
We introduce a natural and more general notion of rectangular density that can be applied to arbitrary cylindric and quasi-polyadic algebras, not just atomic ones. We then show that every rectangularly dense cylindric algebra is representable, and we extend this result to other classes of algebras of logic, for example quasi-polyadic algebras and substitution-cylindrification algebras with and without equality, relation algebras, and special Boolean monoids. The results of op. cit. mentioned above are special cases of our general theorems.
We point out an error in the proof of the Henkin-Monk-Tarski representation theorem for atomic, equality-free, quasi-polyadic algebras with rectangular atoms. The error consists in the implicit assumption of a property that does not, in general, hold. We then give a correct proof of their theorem.
Henkin and Tarski also introduced the notion of a rich cylindric algebra and proved in op. cit. that every rich cylindric algebra of finite dimension (or, more generally, of locally finite dimension) satisfying certain special identities is representable.
We introduce a modification of the notion of a rich algebra that, in our opinion, renders it more natural. In particular, under this modification richness becomes a density notion. Moreover, our notion of richness applies not only to algebras with equality, such as cylindric algebras, but also to algebras without equality. We show that a finite dimensional algebra is rich iff it is rectangularly dense and quasi-atomic; moreover, each of these conditions is also equivalent to a very natural condition of point density. As a consequence, every finite dimensional (or locally finite dimensional) rich algebra of logic is representable. We do not have to assume the validity of any special identities to establish this representability. Not only does this give an improvement of the Henkin-Tarski representation theorem for rich cylindric algebras, it solves positively an open problem in op. cit. concerning the representability of finite dimensional rich quasi-polyadic algebras without equality.

36 citations

••

TL;DR: The axioms of CZF are extended and it is shown that the resulting theory, when augmented by the tertium non-datur, is equivalent to ZF plus the assertion that there are inaccessibles of all transfinite orders.

Abstract: This paper is the first in a series whose objective is to study notions of large sets in the context of formal theories of constructivity. The two theories considered are Aczel's constructive set theory ( CZF ) and Martin-Lof's intuitionistic theory of types. This paper treats Mahlo's π-numbers which give rise classically to the enumerations of inaccessibles of all transfinite orders. We extend the axioms of CZF and show that the resulting theory, when augmented by the tertium non-datur, is equivalent to ZF plus the assertion that there are inaccessibles of all transfinite orders. Finally, the theorems of that extension of CZF are interpreted in an extension of Martin-Lof's intuitionistic theory of types by a universe.

33 citations

••

TL;DR: The attic is gathered to gather miscellaneous results in proof theory from the attic, including an equivalence between transfinite induction rule and iterated reflection schema over IΣn, and proof theoretic strengths of classical fixed points theories.

Abstract: We gather the following miscellaneous results in proof theory from the attic. 1. 1. A provably well-founded elementary ordering admits an elementary order preserving map. 2. 2. A simple proof of an elementary bound for cut elimination in propositional calculus and its applications to separation problem in relativized bounded arithmetic below S21(X). 3. 3. Equivalents for Bar Induction, e.g., reflection schema for ω logic. 4. 4. Direct computations in an equational calculus PRE and a decidability problem for provable inequations in PRE. 5. 5. Intuitionistic fixed point theories which are conservative extensions of HA. 6. 6. Proof theoretic strengths of classical fixed points theories. 7. 7. An equivalence between transfinite induction rule and iterated reflection schema over IΣn. 8. 8. Derivation lengths of finite rewrite rules reducing under lexicographic path orders and multiply recursive functions. Each section can be read separately in principle.

28 citations

••

TL;DR: It is shown that not every (infinitary) first-order theory has a classifying topos in this sense, but those which do by an appropriate ‘smallness condition’ are characterized by the Heyting-valued completeness theorem for infinitary first- order logic.

Abstract: By a classifying topos for a first-order theory T , we mean a topos ∄ such that, for any topos F models of T in F correspond exactly to open geometric morphisms F → E . We show that not every (infinitary) first-order theory has a classifying topos in this sense, but we characterize those which do by an appropriate ‘smallness condition’, and we show that every Grothendieck topos arises as the classifying topos of such a theory. We also show that every first-order theory has a conservative extension to one which possesses a classifying topos, and we obtain a Heyting-valued completeness theorem for infinitary first-order logic.

28 citations

••

TL;DR: A Π01 class is an effectively closed set of reals determined by cardinality, measure and category as well as by the complexity of the members of a class P.

Abstract: A Π01 class is an effectively closed set of reals. We study properties of these classes determined by cardinality, measure and category as well as by the complexity of the members of a class P. Given an effective enumeration {Pe:e

27 citations

••

TL;DR: The Gap Forcing Theorem (GCH) of as discussed by the authors implies that after any reverse Easton iteration of closed forcing, such as the Laver preparation, every supercompactness measure on a super-compact cardinal extends a measure from the ground model.

Abstract: The Gap Forcing Theorem, a key contribution of this paper, implies essentially that after any reverse Easton iteration of closed forcing, such as the Laver preparation, every supercompactness measure on a supercompact cardinal extends a measure from the ground model. Thus, such forcing can create no new supercompact cardinals, and, if the GCH holds, neither can it increase the degree of supercompactness of any cardinal; in particular, it can create no new measurable cardinals. In a crescendo of what I call exact preservation theorems, I use this new technology to perform a kind of partial Laver preparation, and thereby finely control the class of posets which preserve a supercompact cardinal. Eventually, I prove the ‘As You Like It’ Theorem, which asserts that the class of κ -directed closed posets which preserve a supercompact cardinal κ can be made by forcing to conform with any pre-selected local definition which respects the equivalence of forcing. Along the way I separate completely the levels of the superdestructibility hierarchy, and, in an epilogue, prove that the notions of fragility and superdestructibility are orthogonal — all four combinations are possible.

••

TL;DR: It is shown that there exists subsets A and B of the real line which are recursively constructible such that A has a nonrecursive Hausdorff dimension and B has a recursive Hausdorf dimension but has a finite, nonrecursion-based measure.

Abstract: It is shown that there exist subsets A and B of the real line which are recursively constructible such that A has a nonrecursive Hausdorff dimension and B has a recursive Hausdorff dimension (between 0 and 1) but has a finite, nonrecursive Hausdorff measure. It is also shown that there exists a polynomial-time computable curve on the two-dimensional plane that has a nonrecursive Hausdorff dimension between 1 and 2. Computability of Julia sets of computable functions on the real line is investigated. It is shown that there exists a polynomial-time computable function f on the real line whose Julia set is not recurisvely approximable.

••

TL;DR: This work defines Dg A (R) to be the set of Turing degrees of the images f(R) under all isomorphisms f from A to computable models and investigates conditions on A and R which are sufficient and necessary to contain every Turing degree.

Abstract: A model is computable if its domain is a computable set and its relations and functions are uniformly computable. Let A be a computable model and let R be an extra relation on the domain of A . That is, R is not named in the language of A . We define Dg A (R) to be the set of Turing degrees of the images f(R) under all isomorphisms f from A to computable models. We investigate conditions on A and R which are sufficient and necessary for Dg A (R) to contain every Turing degree. These conditions imply that if every Turing degree ⩽ 0″ can be realized in Dg A (R) via an isomorphism of the same Turing degree as its image of R, then Dg A (R) contains every Turing degree. We also discuss an example of A and R whose Dg A (R) coincides with the Turing degrees which are ⩽ 0′.

••

TL;DR: A uniform approach for proving several meta-theorems relating properties of λ-terms and their typability in the systems D and D Ω and characterize the terms that have weak head-normal forms, which appears to be new.

Abstract: It was observed by Curry that when (untyped) λ-terms can be assigned types, for example, simple types, these terms have nice properties (for example, they are strongly normalizing). Coppo, Dezani, and Veneri, introduced type systems using conjunctive types, and showed that several important classes of (untyped) terms can be characterized according to the shape of the types that can be assigned to these terms. For example, the strongly normalizable terms, the normalizable terms, and the terms having head-normal forms, can be characterized in some systems D and D Ω. The proofs use variants of the method of reducibility. In this paper, we present a uniform approach for proving several meta-theorems relating properties of λ-terms and their typability in the systems D and D Ω. Our proofs use a new and more modular version of the reducibility method. As an application of our metatheorems, we show how the characterizations obtained by Coppo, Dezani, Veneri, and Pottinger, can be easily rederived. We also characterize the terms that have weak head-normal forms, which appears to be new. We conclude by stating a number of challenging open problems regarding possible generalizations of the realizability method.

••

TL;DR: This work considers the computably enumerable sets under the relation of Q-reducibility, and uses coding methods to show that the elementary theory of 〈RQ, ⩽Q〉 is undecidable.

Abstract: We consider the computably enumerable sets under the relation of Q-reducibility. We first give several results comparing the upper semilattice of c.e. Q-degrees, 〈RQ, ⩽Q〉, under this reducibility with the more familiar structure of the c.e. Turing degrees. In our final section, we use coding methods to show that the elementary theory of 〈RQ, ⩽Q〉 is undecidable.

••

TL;DR: Cette notion constructive permet une relecture de la theorie d'Artin-Schreier, avec la modification capitale that le resultat final est alors etabli de maniere constructive.

Abstract: Resume Nous introduisons la notion de structure algebrique dynamique, inspiree de l'evaluation dynamique et de la theorie des modeles. Nous montrons comment cette notion constructive permet une relecture de la theorie d'Artin-Schreier, avec la modification capitale que le resultat final est alors etabli de maniere constructive. Nous pensons que ce que nous avons realise ici sur un cas d'ecole peut etre generalise a des parties significatives de l'algebre classique, et est donc une contribution a la realisation du programme de Hilbert pour l'algebre classique.

••

TL;DR: It is proved that if M is an inner model of set theory and the set of reals in M is analytic then either all reals are in $M$ or else $\aleph _1^M$ is countable.

Abstract: We consider the possible complexity of the set of reals belonging to an inner model M of set theory. We show that if this set is analytic then either ℵ1M is countable or else all reals are in M. We also show that if an inner model contains a superperfect set of reals as a subset then it contains all reals. On the other hand, it is possible to have an inner model M whose reals are an uncountable Fσ set and which does not have all reals. A similar construction shows that there can be an inner model M which computes correctly ℵ1, contains a perfect set of reals as a subset and yet not all reals are in M. These results were motivated by questions of H. Friedman and K. Prikry.

••

TL;DR: The numerical strength of fragments of arithmetical comprehension, choice and general uniform boundedness is studied systematically and provably recursive function(al)s of type ⩽2 of the extensions of T n ω are characterized based on these fragments.

Abstract: In this paper the numerical strength of fragments of arithmetical comprehension, choice and general uniform boundedness is studied systematically. These principles are investigated relative to base systems T n ω in all finite types which are suited to formalize substantial parts of analysis but nevertheless have provably recursive function(al)s of low growth. We reduce the use of instances of these principles in T n ω -proofs of a large class of formulas to the use of instances of certain arithmetical principles thereby determining faithfully the arithmetical content of the former. This is achieved using the method of elimination of Skolem functions for monotone formulas which was introduced by the author in a previous paper. As corollaries we obtain new conservation results for fragments of analysis over fragments of arithmetic which strengthen known purely first-order conservation results. We also characterize the provably recursive function(al)s of type ⩽2 of the extensions of T n ω based on these fragments of arithmetical comprehension, choice and uniform boundedness.

••

TL;DR: A first order axiom set is described which yields the classical first order Euclidean geometry of Tarski when used with classical logic, and yields an intuitionistic (or constructive) Euclideans geometry when using with intuitionistic logic.

Abstract: We describe a first order axiom set which yields the classical first order Euclidean geometry of Tarski when used with classical logic, and yields an intuitionistic (or constructive) Euclidean geometry when used with intuitionistic logic. The first order language has a single six place atomic predicate and no function symbols. The intuitionistic system has a computational interpretation in recursive function theory, that is, a realizability interpretation analogous to those given by Kleene for intuitionistic arithmetic and analysis. This interpretation shows the unprovability in the intuitionistic theory of certain “nonconstructive” theorems of the classical geometry.

••

TL;DR: It turns out that the notion of randomness defines, in a natural way, an algebra that is computably enumerable and finitely generated.

Abstract: This paper shows how the notion of randomness defines, in a natural way, an algebra. It turns out that the algebra is computably enumerable and finitely generated. The paper investigates algebraic and effective properties of this algebra.

••

TL;DR: A point-free approach to the study of topological spaces and functions on them is developed, platforms for both are established and some findings on recursive points are presented.

Abstract: In this paper we develop a point-free approach to the study of topological spaces and functions on them, establish platforms for both and present some findings on recursive points. (The effectivization of the functions on our spaces and related results are presented in a sequel.) In the first sections of the paper, we obtain conditions under which our approach leads to the generation of ideal objects (points) with which mathematicians work. Next, we apply the effective version of our approach to the real numbers, and make exact connections to the classical approach to recursive reals. In the succeeding sections of the paper, we introduce machinery to produce functions on topological spaces and find succinct conditions which will be effectivized in our sequel.

••

TL;DR: Two applications of the main Theorem to set theory are presented and a theorem of Magidor's on covering between models of ZFC is proved.

Abstract: The existence of exact upper bounds for increasing sequences of ordinal functions modulo an ideal is discussed. The main theorem (Theorem 18 below) gives a necessary and sufficient condition for the existence of an exact upper bound ƒ for a \ tf = 〈ƒ α : α A where λ > ¦A¦ + is regular: an eub ƒ with lim infI cf ƒ(a) = μ exists if and only if for every regular κ ϵ (¦A¦,μ) the set of flat points in \ tf of cofinality κ is stationary. Two applications of the main Theorem to set theory are presented. A theorem of Magidor's on covering between models of ZFC is proved using the main theorem (Theorem 22): If V⊂-W are transitive models of set theory with ω-covering and GCH holds in V, then κ-covering holds between V and W for all cardinals κ. A new proof of a Theorem by Cummings on collapsing successors of singulars is also given (Theorem 24). The appendix to the paper contains a short proof of Shelah's trichotomy theorem, for the reader's convenience.

••

TL;DR: The nonstandard stroboscopy method is extended to an elementary, yet general class of partial difference/differential equations, both of first and second order, and creates intermediary objects: partial difference equations with S-continuous solutions, which have both discrete and continuous properties.

Abstract: The nonstandard stroboscopy method links discrete-time ordinary difference equations of first-order and continuous-time, ordinary differential equations of first order. We extend this method to the second order, and also to an elementary, yet general class of partial difference/differential equations, both of first and second order. We thus obtain straightforward discretizations and continuizations, even avoiding change of variables. In fact, we create intermediary objects: partial difference equations with S-continuous solutions, which have both discrete and continuous properties. (C) 1998 Elsevier Science B.V. All rights reserved.

••

TL;DR: It is proved that the boundedness theorem of generalized recursion theory can be derived from the ω-completeness theorem for number theory, and a proof of the boundeds theorem which does not refer to the analytical hierarchy theorem.

Abstract: We prove that the boundedness theorem of generalized recursion theory can be derived from the ω-completeness theorem for number theory. This yields a proof of the boundedness theorem which does not refer to the analytical hierarchy theorem.

••

TL;DR: A new kind of genericity is defined, dynamic genericity, and it is proved that it is stronger than pb-genericity, to examine the relation between genericity and array noncomputability and derive some structural information about the Δ20 degrees.

Abstract: We examine notions of genericity intermediate between 1-genericity and 2-genericity, especially in relation to the Δ20 degrees. We define a new kind of genericity, dynamic genericity, and prove that it is stronger than pb-genericity. Specifically, we show there is a Δ20 pb-generic degree below which the pb-generic degrees fail to be downward dense and that pb-generic degrees are downward dense below every dynamically generic degree. To do so, we examine the relation between genericity and array noncomputability, deriving some structural information about the Δ20 degrees in the process.

••

TL;DR: New ∄-definable properties relating the∄-structure of A to deg( A), which answer some open questions are introduced, and a related property NL ( A) is presented which has a slower flow but fast enough to guarantee that A is not low, even though A may possess virtually all other related lowness properties and A may simultaneously be promptly simple.

Abstract: Post in 1944 began studying properties of a computably enumerable (c.e.) set A such as simple, h-simple, and hh-simple, with the intent of finding a property guaranteeing incompleteness of A . From the observations of Post ( 1943 ) and Myhill ( 1956 ), attention focused by the 1950s on properties definable in the inclusion ordering of c.e. subsets of ω, namely E = ( W n neω , ⊂). In the 1950s and 1960s Tennenbaum, Martin, Yates, Sacks, Lachlan, Shoenfield and others produced a number of elegant results relating ∄-definable properties of A , like maximal, hh-simple, atomless, to the information content (usually the Turing degree, deg( A ) of A . Harrington and Soare ( 1991 ) gave an answer to Post's program for definable properties by producing an ∄-definable property Q ( A ) which guarantees that A is incomplete and noncomputable, but developed a new Δ 3 0 -automorphism method to prove certain other properties are not ∄-definable. In this paper we introduce new ∄-definable properties relating the ∄-structure of A to deg( A ), which answer some open questions. In contrast to Q ( A ) we exhibit here an ∄-definable property T ( A ) which allows such a rapid flow of elements into A that A must be complete even though A may possess many other properties such as being promptly simple. We also present a related property NL ( A ) which has a slower flow but fast enough to guarantee that A is not low, even though A may possess virtually all other related lowness properties (low 2 and others) and A may simultaneously be promptly simple.

••

TL;DR: Godel's interpretation of T is extended to theories Pn of “predicative” functionals, which are defined using Martin-Lof's universes of transfinite types, so that each ⌢ID n is interpreted in the corresponding Pn.

Abstract: In 1958 Godel published his Dialectica interpretation, which reduces classical arithmetic to a quantifier-free theory T axiomatizing the primitive recursive functionals of finite type Here we extend Godel's T to theories Pn of “predicative” functionals, which are defined using Martin-Lof's universes of transfinite types We then extend Godel's interpretation to the theories of arithmetic inductive definitions ⌢ID n , so that each ⌢ID n is interpreted in the corresponding Pn Since the strengths of the theories ⌢ID n are cofinal in the ordinal Γ0, as a corollary this analysis provides an ordinal-free characterization of the

••

TL;DR: This work approaches the subject of o-minimality from the point of view of tame systems, following the work of Charbonnel and Wilkie, and is able to obtain the following generalisation of a recent result of Gabrielov.

Abstract: We approach the subject of o-minimality from the point of view of tame systems, following the work of Charbonnel and Wilkie. This gives some general sufficient conditions for a system to be model complete and o-minimal. We are then able to obtain the following generalisation of a recent result of Gabrielov (which in his case applied only to analytic functions): A polynomially bounded o-minimal expansion of the real ordered field by a collection of restricted C∝ functions, which is closed under partial differentiation, is model complete.

••

TL;DR: The method to build notation systems for infinite sequent-style derivations, analogous to well-known systems of notation for ordinals, is extended to the more general setting of infinite (typed) terms in order to make it applicable in other proof-theoretic contexts as well as in recursion theory.

Abstract: Buchholz (1991) presented a method to build notation systems for infinite sequent-style derivations, analogous to well-known systems of notation for ordinals. The essential feature is that from a notation one can read off by a primitive (not e 0 -) recursive function its n th predecessor and, e.g. the last rule applied. Here we extend the method to the more general setting of infinite (typed) terms, in order to make it applicable in other proof-theoretic contexts as well as in recursion theory. As examples, we use the method to 1. (1) give a new proof of a well-known trade-off theorem (Schwichtenberg, 1975), which says that detours through higher types can be eliminated by the use of transfinite recursion along higher ordinals, and 2. (2) construct a continuous normalization operator with an explicit modulus of continuity.

••

TL;DR: The main result shows that any computable theory T of intuitionistic predicate logic has a Kripke model with decidable forcing such that for any sentence φ, φ is forced in the model if and only ifπ is intuitionistically deducible from T.

Abstract: In this paper we introduce effectiveness into model theory of intuitionistic logic. The main result shows that any computable theory T of intuitionistic predicate logic has a Kripke model with decidable forcing such that for any sentence φ, φ is forced in the model if and only if φ is intuitionistically deducible from T .