The main result proved in the paper is that on every recursive structure the intrinsically hyperarithmetical sets coincide with the relatively intrinsically hyperarithmetical sets. As a side effect of the proof an effective version of the Kueker's theorem on definability by means of infinitary formulas is obtained.

We prove that there is a structure, indeed a linear ordering, whose degree spectrum is the set of all non-hyperarithmetic degrees. We also show that degree spectra can distinguish measure from category.

Millar showed that for each n<ω, there is a complete decidable theory having precisely eighteen nonisomorphic countable models where some of these are decidable exactly in the hyperarithmetic set H. By combining ideas from Millar's proof with a technique of Peretyat'kin, the author reduces the number of countable models to five. By a theorem of Millar, this is the smallest number of countable models a decidable theory can have if some of the models are not 0″-decidable.

A statement of hyperarithmetic analysis is a sentence of second order arithmetic S such that for every Y⊆ω, the minimum ω-model containing Y of RCA0 + S is HYP, the ω-model consisting of the sets hyperarithmetic in Y. We provide an example of a mathematical theorem which is a statement of hyperarithmetic analysis. This statement, that we call INDEC, is due to Jullien [13]. To the author's knowledge, no other already published, purely mathematical statement has been (...) found with this property until now. We also prove that, over RCA0, INDEC is implied by [Formula: see text] and implies ACA0, but of course, neither ACA0, nor ACA 0+ imply it. We introduce five other statements of hyperarithmetic analysis and study the relations among them. Four of them are related to finitely-terminating games. The fifth one, related to iterations of the Turing jump, is strictly weaker than all the other statements that we study in this paper, as we prove using Steel's method of forcing with tagged trees. (shrink)

A mass problem is a set of Turing oracles. If P and Q are mass problems, we say that P is weakly reducible to Q if for all Y ∈ Q there exists X ∈ P such that X is Turing reducible to Y. A weak degree is an equivalence class of mass problems under mutual weak reducibility. Let [Formula: see text] be the lattice of weak degrees of mass problems associated with nonempty [Formula: see text] subsets of the Cantor (...) space. The lattice [Formula: see text] has been studied in previous publications. The purpose of this paper is to show that [Formula: see text] partakes of hyperarithmeticity. We exhibit a family of specific, natural degrees in [Formula: see text] which are indexed by the ordinal numbers less than [Formula: see text] and which correspond to the hyperarithmetical hierarchy. Namely, for each [Formula: see text], let hα be the weak degree of 0, the αth Turing jump of 0. If p is the weak degree of any mass problem P, let p* be the weak degree of the mass problem P* = {Y | ∃X ⊆ BLR )} where BLR is the set of functions which are boundedly limit recursive in X. Let 1 be the top degree in [Formula: see text]. We prove that all of the weak degrees [Formula: see text], [Formula: see text], are distinct and belong to [Formula: see text]. In addition, we prove that certain index sets associated with [Formula: see text] are [Formula: see text] complete. (shrink)

A generic computation of a subsetAof ℕ is a computation which correctly computes most of the bits ofA, but which potentially does not halt on all inputs. The motivation for this concept is derived from complexity theory, where it has been noticed that frequently, it is more important to know how difficult a type of problem is in the general case than how difficult it is in the worst case. When we study this concept from a recursion theoretic point of (...) view, to create a transitive relationship, we are forced to consider oracles that sometimes fail to give answers when asked questions. Unfortunately, this makes working in the generic degrees quite difficult. Indeed, we show that generic reduction is$\Pi _1^1$―complete. To help avoid this difficulty, we work with the generic degrees of density-1 reals. We demonstrate how an understanding of these degrees leads to a greater understanding of the overall structure of the generic degrees, and we also use these density-1 sets to provide a new a characterization of the hyperartithmetical Turing degrees. (shrink)

We study arithmetic and hyperarithmetic degrees of categoricity. We extend a result of E. Fokina, I. Kalimullin, and R. Miller to show that for every computable ordinal $\alpha$, $\mathbf{0}^{}$ is the degree of categoricity of some computable structure $\mathcal{A}$. We show additionally that for $\alpha$ a computable successor ordinal, every degree $2$-c.e. in and above $\mathbf{0}^{}$ is a degree of categoricity. We further prove that every degree of categoricity is hyperarithmetic and show that the index set of structures (...) with degrees of categoricity is $\Pi_{1}^{1}$-complete. (shrink)

We study randomness beyond${\rm{\Pi }}_1^1$-randomness and its Martin-Löf type variant, which was introduced in [16] and further studied in [3]. Here we focus on a class strictly between${\rm{\Pi }}_1^1$and${\rm{\Sigma }}_2^1$that is given by the infinite time Turing machines introduced by Hamkins and Kidder. The main results show that the randomness notions associated with this class have several desirable properties, which resemble those of classical random notions such as Martin-Löf randomness and randomness notions defined via effective descriptive set theory such as${\rm{\Pi (...) }}_1^1$-randomness. For instance, mutual randoms do not share information and a version of van Lambalgen’s theorem holds.Towards these results, we prove the following analogue to a theorem of Sacks. If a real is infinite time Turing computable relative to all reals in some given set of reals with positive Lebesgue measure, then it is already infinite time Turing computable. As a technical tool towards this result, we prove facts of independent interest about random forcing over increasing unions of admissible sets, which allow efficient proofs of some classical results about hyperarithmetic sets. (shrink)

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 < ω} of the Π01 classes, the index set I for a certain property is the set of indices e such that Pe has the property. For example, the index set of binary Π01 classes of positive measure is Σ02 complete. (...) Various notions of boundedness are discussed and classified. For example, the index set of the recursively bounded classes is Σ03 complete and the index set of the recursively bounded classes which have infinitely many recursive members is Π04 complete. Consideration of the Cantor-Bendixson derivative leads to index sets in the transfinite levels of the hyperarithmetic hierarchy. (shrink)

The infinite pigeonhole principle for 2-partitions asserts the existence, for every set A, of an infinite subset of A or of its complement. In this paper, we study the infinite pigeonhole pr...

Solovay has shown that if $\cal{O}$ is an open subset of $P(\omega)$ with code $S$ and no infinite set avoids $\cal{O}$ , then there is an infinite set hyperarithmetic in $S$ that lands in $\cal{O}$ . We provide a direct proof of this theorem that is easily formalizable in $ATR_0$.

We give a new characterization of the hyperarithmetic sets: a set X of integers is recursive in e α if and only if there is a Turing machine which computes X and "halts" in less than or equal to the ordinal number ω α of steps. This result represents a generalization of the well-known "limit lemma" due to J. R. Shoenfield [Sho-1] and later independently by H. Putnam [Pu] and independently by E. M. Gold [Go]. As an application (...) of this result, we give a recursion theoretic analysis of clopen determinacy: there is a correlation given between the height α of a well-founded tree corresponding to a clopen game $A \subseteq \omega^\omega$ and the Turing degree of a winning strategy f for one of the players--roughly, f can be chosen to be recursive in 0 α and this is the best possible (see § 4 for precise results). (shrink)

This paper continues joint work of the authors with P. Clote, R. Soare and S. Wainer (Annals of Pure and Applied Logic, vol. 31 (1986), pp. 145--163). An element x of the Cantor space 2 ω is said have rank α in the closed set P if x is in $D^\alpha(P)\backslash D^{\alpha + 1}(P)$ , where D α is the iterated Cantor-Bendixson derivative. The rank of x is defined to be the least α such that x has rank α in (...) some Π 0 1 set. The main result of the five-author paper is that for any recursive ordinal λ + n (where λ is a limit and n is finite), there is a point with rank λ + n which is Turing equivalent to O (λ + 2n) . All ranked points constructed in that paper are Π 0 2 singletons. We now construct a ranked point which is not a Π 0 2 singleton. In the previous paper the points of high rank were also of high hyperarithmetic degree. We now construct ▵ 0 2 points with arbitrarily high rank. We also show that every nonrecursive RE point is Turing equivalent to an RE point of rank one and that every nonrecursive ▵ 0 2 point is Turing equivalent to a hyperimmune point of rank one. We relate Clote's notion of the height of a Π 0 1 singleton in the Baire space with the notion of rank. Finally, we show that every hyperimmune point x is Turing equivalent to a point which is not ranked. (shrink)

In (Fund Math 60:175–186 1967), Wolk proved that every well partial order (wpo) has a maximal chain; that is a chain of maximal order type. (Note that all chains in a wpo are well-ordered.) We prove that such maximal chain cannot be found computably, not even hyperarithmetically: No hyperarithmetic set can compute maximal chains in all computable wpos. However, we prove that almost every set, in the sense of category, can compute maximal chains in all computable wpos. Wolk’s original (...) result actually shows that every wpo has a strongly maximal chain, which we define below. We show that a set computes strongly maximal chains in all computable wpo if and only if it computes all hyperarithmetic sets. (shrink)

We study the structure of Σ11 equivalence relations on hyperarithmetical subsets of ω under reducibilities given by hyperarithmetical or computable functions, called h-reducibility and FF-reducibility, respectively. We show that the structure is rich even when one fixes the number of properly equation imagei.e., Σ11 but not equation image equivalence classes. We also show the existence of incomparable Σ11 equivalence relations that are complete as subsets of ω × ω with respect to the corresponding reducibility on sets. We study complete (...) Σ11 equivalence relations and show that existence of infinitely many properly Σ11 equivalence classes that are complete as Σ11 sets is necessary but not sufficient for a relation to be complete in the context of Σ11 equivalence relations. (shrink)

This book is a contribution to the flourishing field of formal and philosophical work on truth and the semantic paradoxes. Our aim is to present several theories of truth, to investigate some of their model-theoretic, recursion-theoretic and proof-theoretic aspects, and to evaluate their philosophical significance. In Part I we first outline some motivations for studying formal theories of truth, fix some terminology, provide some background on Tarski’s and Kripke’s theories of truth, and then discuss the prospects of classical type-free truth. (...) In Chapter 4 we discuss some minimal adequacy conditions on a satisfactory theory of truth based on the function that the truth predicate is intended to fulfil on the deflationist account. We cast doubt on the adequacy of some non-classical theories of truth and argue in favor of classical theories of truth. Part II is devoted to grounded truth. In chapter 5 we introduce a game-theoretic semantics for Kripke’s theory of truth. Strategies in these games can be interpreted as reference-graphs of the sentences in question. Using that framework, we give a graph-theoretic analysis of the Kripke-paradoxical sentences. In chapter 6 we provide simultaneous axiomatizations of groundedness and truth, and analyze the proof-theoretic strength of the resulting theories. These range from conservative extensions of Peano arithmetic to theories that have the full strength of the impredicative system ID1. Part III investigates the relationship between truth and set-theoretic comprehen- sion. In chapter 7 we canonically associate extensions of the truth predicate with Henkin-models of second-order arithmetic. This relationship will be employed to determine the recursion-theoretic complexity of several theories of grounded truth and to show the consistency of the latter with principles of generalized induction. In chapter 8 it is shown that the sets definable over the standard model of the Tarskian hierarchy are exactly the hyperarithmetical sets. Finally, we try to apply a certain solution to the set-theoretic paradoxes to the case of truth, namely Quine’s idea of stratification. This will yield classical disquotational theories that interpret full second-order arithmetic without set parameters, Z2- . We also indicate a method to recover the parameters. An appendix provides some background on ordinal notations, recursion theory and graph theory. (shrink)

Solovay has shown that if F: [ω] ω → 2 is a clopen partition with recursive code, then there is an infinite homogeneous hyperarithmetic set for the partition (a basis result). Simpson has shown that for every 0 α , where α is a recursive ordinal, there is a clopen partition F: [ω] ω → 2 such that every infinite homogeneous set is Turing above 0 α (an anti-basis result). Here we refine these results, by associating the "order type" (...) of a clopen set with the Turing complexity of the infinite homogeneous sets. We also consider the Nash-Williams barrier theorem and its relation to the clopen Ramsey theorem. (shrink)

We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, $\scr{D}_{h}$ . In fact, we prove that every sublattice of any hyperarithmetic lattice (and so, in particular, every countable, locally finite lattice) is isomorphic to an initial segment of $\scr{D}_{h}$ . Corollaries include the decidability of the two quantifier theory of $\scr{D}_{h}$ and the undecidability of its three quantifier theory. The key tool in the proof is (...) a new lattice representation theorem that provides a notion of forcing for which we can prove a version of the fusion lemma in the hyperarithmetic setting and so the preservation of $\omega _{1}^{CK}$ . Somewhat surprisingly, the set theoretic analog of this forcing does not preserve ω₁. On the other hand, we construct countable lattices that are not isomorphic to any initial segment of $\scr{D}_{h}$. (shrink)

We characterize the completely determined Borel subsets of HYP as exactly the [Formula: see text] subsets of HYP. As a result, HYP believes there is a Borel well-ordering of the reals, that the Borel Dual Ramsey Theorem fails, and that every Borel d-regular bipartite graph has a Borel perfect matching, among other examples. Therefore, the Borel Dual Ramsey Theorem and several theorems of descriptive combinatorics are not theories of hyperarithmetic analysis. In the case of the Borel Dual Ramsey Theorem, (...) this answers a question of Astor, Dzhafarov, Montalbán, Solomon and the third author. (shrink)

An external characterization of the inductive sets on countable abstract structures is presented. The main result is an abstract version of the classical Suslin-Kleene characterization of the hyperarithmetical sets.

We consider implicit definability over the natural number system $\mathbb{N},+,\times,=$. We present a new proof of two theorems of Leo Harrington. The first theorem says that there exist implicitly definable subsets of $\mathbb{N}$ which are not explicitly definable from each other. The second theorem says that there exists a subset of $\mathbb{N}$ which is not implicitly definable but belongs to a countable, explicitly definable set of subsets of $\mathbb{N}$. Previous proofs of these theorems have used finite- or infinite-injury priority constructions. (...) Our new proof is easier in that it uses only a nonpriority oracle construction, adapted from the standard proof of the Friedberg jump theorem. (shrink)

In this paper, we give an example of a complete computable infinitary theory T with countable models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ and T has no uncountable model. In fact, ${\mathcal{M}}$ and ${\mathcal{N}}$ are (up to isomorphism) the only models of T. Moreover, for all computable ordinals α, the computable ${\Sigma_\alpha}$ part of T is hyperarithmetical. It follows from a theorem of Gregory (JSL 38:460–470, 1972; Not Am Math Soc 17:967–968, 1970) (...) that if T is a Π 1 1 set of computable infinitary sentences and T has a pair of models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ , then T would have an uncountable model. (shrink)

A set of integers A is computably encodable if every infinite set of integers has an infinite subset computing A. By a result of Solovay, the computably encodable sets are exactly the hyperarithmetic ones. In this article, we extend this notion of computable encodability to subsets of the Baire space, and we characterize the Π10-encodable compact sets as those which admit a nonempty Σ11-subset. Thanks to this equivalence, we prove that weak weak König’s lemma is not strongly (...) computably reducible to Ramsey’s theorem. This answers a question of Hirschfeldt and Jockusch. (shrink)

The Friedman–Stanley jump, extensively studied by descriptive set theorists, is a fundamental tool for gauging the complexity of Borel isomorphism relations. This paper focuses on a natural computable analog of this jump operator for equivalence relations on $\omega $, written ${\dotplus }$, recently introduced by Clemens, Coskey, and Krakoff. We offer a thorough analysis of the computable Friedman–Stanley jump and its connections with the hierarchy of countable equivalence relations under the computable reducibility $\leq _c$. In particular, we show that this (...) jump gives benchmark equivalence relations going up the hyperarithmetic hierarchy and we unveil the complicated highness hierarchy that arises from ${\dotplus }$. (shrink)

Where AR is the set of arithmetic Turing degrees, 0 (ω ) is the least member of { $\mathbf{\alpha}^{(2)}|\mathbf{a}$ is an upper bound on AR}. This situation is quite different if we examine HYP, the set of hyperarithmetic degrees. We shall prove (Corollary 1) that there is an a, an upper bound on HYP, whose hyperjump is the degree of Kleene's O. This paper generalizes this example, using an iteration of the jump operation into the transfinite which is based (...) on results of Jensen and is detailed in [3] and [4]. In $\S1$ we review the basic definitions from [3] which are needed to state the general results. (shrink)

The Hanf number for a set S of sentences in \ is the least infinite cardinal \ such that for all \, if \ has models in all infinite cardinalities less than \, then it has models of all infinite cardinalities. Friedman asked what is the Hanf number for Scott sentences of computable structures. We show that the value is \. The same argument proves that \ is the Hanf number for Scott sentences of hyperarithmetical structures.

Martin’s conjecture is an attempt to classify the behavior of all definable functions on the Turing degrees under strong set theoretic hypotheses. Very roughly it says that every such function is either eventually constant, eventually equal to the identity function or eventually equal to a transfinite iterate of the Turing jump. It is typically divided into two parts: the first part states that every function is either eventually constant or eventually above the identity function and the second part states that (...) every function which is above the identity is eventually equal to a transfinite iterate of the jump. If true, it would provide an explanation for the unique role of the Turing jump in computability theory and rule out many types of constructions on the Turing degrees.In this thesis, we will introduce a few tools which we use to prove several cases of Martin’s conjecture. It turns out that both these tools and these results on Martin’s conjecture have some interesting consequences both for Martin’s conjecture and for a few related topics.The main tool that we introduce is a basis theorem for perfect sets, improving a theorem due to Groszek and Slaman. We also introduce a general framework for proving certain special cases of Martin’s conjecture which unifies a few pre-existing proofs. We will use these tools to prove three main results about Martin’s conjecture: that it holds for regressive functions on the hyperarithmetic degrees, that part 1 holds for order preserving functions on the Turing degrees, and that part 1 holds for a class of functions that we introduce, called measure preserving functions.This last result has several interesting consequences for the study of Martin’s conjecture. In particular, it shows that part 1 of Martin’s conjecture is equivalent to a statement about the Rudin-Keisler order on ultrafilters on the Turing degrees. This suggests several possible strategies for working on part 1 of Martin’s conjecture, which we will discuss.The basis theorem that we use to prove these results also has some applications outside of Martin’s conjecture. We will use it to prove a few theorems related to Sacks’ question about whether it is provable in $\mathsf {ZFC}$ that every locally countable partial order of size continuum embeds into the Turing degrees. We will show that in a certain extension of $\mathsf {ZF}$, this holds for all partial orders of height two, but not for all partial orders of height three. Our proof also yields an analogous result for Borel partial orders and Borel embeddings in $\mathsf {ZF}$, which shows that the Borel version of Sacks’ question has a negative answer.We will end the thesis with a list of open questions related to Martin’s conjecture, which we hope will stimulate further research.prepared by Patrick Lutz.E-mail: [email protected]. (shrink)

Recall that [Formula: see text] is the lattice of Muchnik degrees of nonempty effectively compact sets in Euclidean space. We solve a long-standing open problem by proving that [Formula: see text] is dense, i.e. satisfies [Formula: see text]. Our proof combines an oracle construction with hyperarithmetical theory.

While finitely generic dimension groups are known to admit no proper self-embeddings, these groups also have no automorphisms other than scalar multiplications, and every countable infinitely generic dimension group admits proper self-embeddings and has automorphisms other than scalar multiplications. The finite-forcing companion of the theory of dimension groups is recursively isomorphic to first-order arithmetic, the infinite-forcing companion of the theory of dimension groups is recursively isomorphic to second-order arithmetic, and the first-order theory of existentially closed dimension groups is a complete (...) $\Pi_{1}^{1}$-set. While many special properties of f.g. dimension groups may be realized in recursive e.c. dimension groups, and many special properties of i.g. dimension groups may be realized in hyperarithmetic e.c. dimension groups, no f.g. dimension group is arithmetic and no i.g. dimension group is analytical. Yet there is an f.g. dimension group recursive in first-order arithmetic, and there is an i.g. dimension group recursive in second-order arithmetic. (shrink)

A well known fact is that every Lebesgue measurable set is regular, i.e., it includes an F$_{\sigma}$ set of the same measure. We analyze this fact from a metamathematical or foundational standpoint. We study a family of Muchnik degrees corresponding to measure-theoretic regularity at all levels of the effective Borel hierarchy. We prove some new results concerning Nies's notion of LR-reducibility. We build some $\omega$-models of RCA$_0$which are relevant for the reverse mathematics of measure-theoretic regularity.

We exploit properties of certain directed graphs, obtained from the families of sets with special effective enumeration properties, to generalize several results in computable model theory to higher levels of the hyperarithmetical hierarchy. Families of sets with such enumeration features were previously built by Selivanov, Goncharov, and Wehner. For a computable successor ordinal α, we transform a countable directed graph into a structure such that has a isomorphic copy if and only if has a computable isomorphic copy.A computable (...) structure is categorical if for all computable isomorphic copies of , there is an isomorphism from onto , which is . We prove that for every computable successor ordinal α, there is a computable, categorical structure, which is not relatively categorical. This generalizes the result of Goncharov that there is a computable, computably categorical structure, which is not relatively computably categorical.An additional relation R on the domain of a computable structure is intrinsically on if in all computable isomorphic copies of , the image of R is . We prove that for every computable successor ordinal α, there is an intrinsically relation on a computable structure, which not relatively intrinsically . This generalizes the result of Manasse that there is an intrinsically computably enumerable relation on a computable structure, which is not relatively intrinsically computably enumerable.The dimension of a structure is the number of computable isomorphic copies, up to isomorphisms. We prove that for every computable successor ordinal α and every n≥1, there is a computable structure with dimension n. This generalizes the result of Goncharov that there is a structure of computable dimension n for every n≥1.Finally, we prove that for every computable successor ordinal α, there is a countable structure with isomorphic copies in just the Turing degrees of sets X such that relative to X is not . In particular, for every finite n, there is a structure with isomorphic copies in exactly the non- Turing degrees. This generalizes the result obtained by Wehner, and independently by Slaman, that there is a structure with isomorphic copies in exactly the nonzero Turing degrees. (shrink)

We prove that, for each countable ordinal ξ≥1, there exist some -complete ω-powers, and some -complete ω-powers, extending previous works on the topological complexity of ω-powers [O. Finkel, Topological properties of omega context free languages, Theoretical Computer Science 262 669–697; O. Finkel, Borel hierarchy and omega context free languages, Theoretical Computer Science 290 1385–1405; O. Finkel, An omega-power of a finitary language which is a borel set of infinite rank, Fundamenta informaticae 62 333–342; D. Lecomte, Sur les ensembles de phrases (...) infinies constructibles a partir d’un dictionnaire sur un alphabet fini, Séminaire d’Initiation a l’Analyse, 1, année 2001–2002; D. Lecomte, Omega-powers and descriptive set theory, Journal of Symbolic Logic 70 1210–1232; J. Duparc, O. Finkel, An ω-Power of a Context-Free Language Which Is Borel Above , in: S. Bold, B. Löwe, T. Räsch, J. van Benthem , in the Proceedings of the international conference foundations of the formal sciences V : Infinite Games, November 26th to 29th, 2004, Bonn, Germany, in: Studies in Logic, vol. 11, College Publications at King’s College, 2007, pp. 109–122]. We prove effective versions of these results; in particular, for each recursive ordinal there exist some recursive sets A2<ω such that , where and denote classes of the hyperarithmetical hierarchy. To do this, we prove effective versions of a result by Kuratowski, describing a set as the range of a closed subset of the Baire space ωω by a continuous bijection. This leads us to prove closure properties for the pointclasses in arbitrary recursively presented Polish spaces. We apply our existence results to get better computations of the topological complexity of some sets of dictionaries considered in [D. Lecomte, Omega-powers and descriptive set theory, Journal of Symbolic Logic 70 1210–1232]. (shrink)

We define the notion of a completely determined Borel code in reverse mathematics, and consider the principle $CD - PB$, which states that every completely determined Borel set has the property of Baire. We show that this principle is strictly weaker than $AT{R_0}$. Any ω-model of $CD - PB$ must be closed under hyperarithmetic reduction, but $CD - PB$ is not a theory of hyperarithmetic analysis. We show that whenever $M \subseteq {2^\omega }$ is the second-order part of (...) an ω-model of $CD - PB$, then for every $Z \in M$, there is a $G \in M$ such that G is ${\rm{\Delta }}_1^1$-generic relative to Z. (shrink)

Computability theory is the mathematical theory of algorithms, which explores the power and limitations of computation. Classical computability theory formalized the intuitive notion of an algorithm and provided a theoretical basis for digital computers. It also demonstrated the limitations of algorithms and showed that most sets of natural numbers and the problems they encode are not decidable (Turing computable). Important results of modern computability theory include the classification of the computational difficulty of sets and problems. Arithmetical and hyperarithmetical (...) hierarchies and the Turing degrees provide important measures of the level of such difficulty. There are uncountably many Turing degrees and they are partially ordered. These classifications are used in contemporary research to investigate the complexity of problems in modern computable mathematics, for example, regarding structures and their properties. (shrink)

Louveau showed that if a Borel set in a Polish space happens to be in a Borel Wadge class $$\Gamma $$, then its $$\Gamma $$ -code can be obtained from its Borel code in a hyperarithmetical manner. We extend Louveau’s theorem to Borel functions: If a Borel function on a Polish space happens to be a $$ \underset{\widetilde{}}{\varvec{\Sigma }}\hbox {}_t$$ -function, then one can find its $$ \underset{\widetilde{}}{\varvec{\Sigma }}\hbox {}_t$$ -code hyperarithmetically relative to its Borel code. More generally, we prove (...) extension-type, domination-type, and decomposition-type variants of Louveau’s theorem for Borel functions. (shrink)

Mancosu, P., Generalizing classical and effective model theory in theories of operations and classes, Annas of Pure and Applied Logic 52 249-308 . In this paper I propose a family of theories of operations and classes with the aim of developing abstract versions of model-theoretic results. The systems are closely related to those introduced and already used by Feferman for developing his program of ‘explicit mathematics’. The theories in question are two-sorted, with one kind of variable for individuals and the (...) other for classes. The individual variables range over a domain closed under pairing and containing, among other things, natural numbers and operations.All the theories used assume a common group of axioms that insure that the individuals in the domain satisfy the conditions for a partial combinatory algebra with pairing. We also assume a class of natural numbers and a class induction axiom on . Finally there are various class existence axioms. I work mainly with three theories, FMT0, FMT and FMTΩ, which differ from each other both in their class existence axioms as well as in some further applicative axioms.These theories have various interpretations in which every object is explicitly presented by some means of definition as well as classical, or ‘standard’, set-theoretical interpretation. The systems are formulated in a flexible general language which can be used to prove abstract theorems of mathematics , thereby generalizing recursive, hyperarithmetic and admissible versions of classical mathematics.The aim of my work is to give an abstract development of portions of model theory in the afore-mentioned theories, in a way to look as much as possible like classical mathematics. Thus, FMT0 generalizes portions of countable and recursive model theory; FMT further generalizes portions of countable and hyperarithmetical model theory and finally FMTΩ provides a generalization of the classical L-completeness theorem and of an admissible version of L-completeness due to Bruce and Keisler. This continues the development of the program mentioned above, originating with Feferman. (shrink)

We present and analyze \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F_\sigma}$$\end{document}-Mathias forcing, which is similar but tamer than Mathias forcing. In particular, we show that this forcing preserves certain weak subsystems of second-order arithmetic such as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{ACA}_0}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathsf{WKL}_0 + \mathsf{I}\Sigma^0_2}$$\end{document}, whereas Mathias forcing does not. We also show that the needed reals for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} (...) \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F_\sigma}$$\end{document}-Mathias forcing (in the sense of Blass in Ann Pure Appl Logic 109(1–2):77–88, 2001) are just the computable reals, as opposed to the hyperarithmetic reals for Mathias forcing. (shrink)

O objetivo deste artigo é apresentar a analogia básica entre arquitetura e retórica antiga a partir dos tratados De Architectura, de Vitrúvio, e o De Oratore, de Cícero. A analogia se verifica na definição do artífice, dos gêneros e partes das técnicas e dos fins de cada uma delas. Para tanto, tomaram-se como referência as fontes do tratado vitruviano, que menciona a influência de Varrão na gramática, de Lucrécio na filosofia e de Cícero no método oratório. A analogia com Cícero (...) serve tanto para conferir autoridade ao modelo de arquiteto vitruviano quanto para diferenciar a arquitetura das demais ciências, sobretudo a filosofia, objeto de debate na obra ciceroniana. (shrink)

Study, with text of the Yogasūtra of Patañjali, text on Yoga philosophy.