Sacks [2] has asked whether there exists a uniform solution to Post's problem, i.e. an enumeration operation W such that $\mathbf{d} for every degree d. It is shown here that if such an operation W exists it cannot itself in a particular technical sense be uniform. In fact, the jump operation is characterized amongst such uniform enumeration operations by the condition: $\mathbf{d} for all d. In addition, it is proved that the only other uniform enumeration operations such (...) that d ≤ W (d) for all d are those which equal the identity operation above some fixed degrees. (shrink)

In the present paper we concentrate on fundamental problems concerning ω-operations over partial enumerated sets. The notion of “HOM-lifts” seems to be an adequate tool for this kind of investigations. MSC: 03D45, 18A30.

The jump operator on the ω-enumeration degrees was introduced in [I.N. Soskov, The ω-enumeration degrees, J. Logic Computat. 17 1193–1214]. In the present paper we prove a jump inversion theorem which allows us to show that the enumeration degrees are first order definable in the structure of the ω-enumeration degrees augmented by the jump operator. Further on we show that the groups of the automorphisms of and of the enumeration degrees are isomorphic. In the second (...) part of the paper we study the jumps of the ω-enumeration degrees below . We define the ideal of the almost zero degrees and obtain a natural characterization of the class H of the ω-enumeration degrees below which are high n for some n and of the class L of the ω-enumeration degrees below which are low n for some n. (shrink)

We show that the e-degree 0'e and the map u ↦ u' are definable in the upper semilattice of all e-degrees. The class of total e-degrees ≥0'e is also definable.

In the present paper, we show the first-order definability of the jump operator in the upper semi-lattice of the \-enumeration degrees. As a consequence, we derive the isomorphicity of the automorphism groups of the enumeration and the \-enumeration degrees.

We prove that there exists a noncappable enumeration degree strictly below 0' e.

We discuss a notion of forcing that characterizes enumeration 1-genericity, and we investigate the immunity, lowness, and quasiminimality properties of enumeration 1-generic sets and their degrees. We construct an enumeration operator Δ such that, for any A, the set ΔA is enumeration 1-generic and has the same jump complexity as A. We deduce from this and other recent results from the literature that not only does every degree a bound an enumeration 1-generic degree b such (...) that a'=b', but also that, if a is nonzero, then we can find such b satisfying 0eenumeration 1-generic degree, hence proving that the class of 1-generic degrees is properly subsumed by the class of enumeration 1-generic degrees. (shrink)

Using only propositional connectives and the provability predicate of a Σ1-sound theory T containing Peano Arithmetic we define recursively enumerable relations that are complete for specific natural classes of relations, as the class of all r. e. relations, and the class of all strict partial orders. We apply these results to give representations of these classes in T by means of formulas.

Enumeration reducibility is a notion of relative computability between sets of natural numbers where only positive information about the sets is used or produced. Extending e‐reducibility to partial functions characterises relative computability between partial functions. We define a polynomial time enumeration reducibility that retains the character of enumeration reducibility and show that it is equivalent to conjunctive non‐deterministic polynomial time reducibility. We define the polynomial time e‐degrees as the equivalence classes under this reducibility and investigate their structure (...) on the recursive sets, showing in particular that the pe‐degrees of the computable sets are dense and do not form a lattice, but that minimal pairs exist. We define a jump operator and use it to produce a characterisation of the polynomial hierarchy. (shrink)

A set A is symmetric enumeration (se-) reducible to a set B (A ≤\sb se B) if A is enumeration reducible to B and \barA is enumeration reducible to \barB. This reducibility gives rise to a degree structure (D\sb se) whose least element is the class of computable sets. We give a classification of ≤\sb se in terms of other standard reducibilities and we show that the natural embedding of the Turing degrees (D\sb T) into the (...) class='Hi'>enumeration degrees (D\sb e) translates to an embedding (ι\sb se) into D\sb se that preserves least element, suprema, and infima. We define a weak and a strong jump and we observe that ι\sb se preserves the jump operator relative to the latter definition. We prove various (global) results concerning branching, exact pairs, minimal covers, and diamond embeddings in D\sb se. We show that certain classes of se-degrees are first-order definable, in particular, the classes of semirecursive, Σ\sb n ⋃ Π\sb n, Δ\sb n (for any n \in ω), and embedded Turing degrees. This last result allows us to conclude that the theory of D\sb se has the same 1-degree as the theory of Second-Order Arithmetic. (shrink)

Let N, O and S denote the set of nonnegative integers, the graph of the constant 0 function and the graph of the successor function respectively. For sets $P, Q, R \subseteq N^2$ operations of transposition, composition, and bracketing are defined as follows: $P^\cup = \{\langle x, y\rangle | \langle y, x\rangle \epsilon P\}, PQ = \{\langle x, z\rangle| \exists y\langle x, y\rangle \epsilon P & \langle y, z\rangle \epsilon Q\}$ , and [ P, Q, R] = ∪n ε M(PnQR (...) n). THEOREM. The class of recursively enumerable subsets of N2 is the smallest class of sets with O and S as members and closed under transposition, composition, and bracketing. This result is derived from a characterization by Julia Robinson of the class of general recursive functions of one variable in terms of function composition and "definition by general recursion." A key step in the proof is to show that if a function F is defined by general recursion from functions A, M, P and R then F = [ P∪, A∪ M, R]. The above definitions of the transposition, composition, and bracketing operations on subsets of N2 can be generalized to subsets of X2 for an arbitrary set X. In this abstract setting it is possible to show that the bracket operation can be defined in terms of K, L, transposition, composition, intersection, and reflexive transitive closure where K: X → X and L: X → X are functions for decoding pairs. (shrink)

One of the main problems in effective model theory is to find an appropriate information complexity measure of the algebraic structures in the sense of computability. Unlike the commonly used degrees of structures, the structure degree measure is total. We introduce and study the jump operation for structure degrees. We prove that it has all natural jump properties (including jump inversion theorem, theorem of Ash), which show that our definition is relevant. We study the relation between the structure degree jump (...) (in the sense of Soskov) and the jump degrees of a structure (in the sense of Jockusch) and give necessary and sufficient conditions for their existence in the terms of structure degrees. We show some properties, distinguishing the structure degrees from the enumeration degrees. (shrink)

This paper is dedicated to the study of properties of the operations ∪ and ∩ in the upper semilattice of the e-degrees as well as in the interval (c,c') e for any e-degree c.

We define an applicative theoryCL 2 similar to combinatory logic which can be interpreted in classes of functions possessing an enumerating function. In contrast to the models of classical combinatory logic, it is not necessarily assumed that the enumerating function itself belongs to that function class. Thereby we get a variety of possible models including e. g. the classes of primitive recursive, recursive, elementary, polynomial-time comptable ofɛ 0-recursive functions.We show that inCL 2 a major part of the metatheory of enumerated (...) classes of functions can be developed. Namely, a kind of λ-abstraction can be defined and abstract versions of theS n m - and (Primitive) Recursion Theorems are proved. Thereby, a closer analysis of the phenomenon of the different recursion theorems is achieved.A theory closely related toCL 2 can be used to replace the applicative part of Feferman's theories for explicit mathematics. So this work can be seen as an answer to Feferman's question to formulate a theory for explicit mathematics in which operations can be interpreted as primitive recursive or even more feasible ones.Finally it is shown that the proof-theoretical strength of various theoreies for explicit mathematics is preserved when replacing the applicative part of the theories by our theory together with an operation for primitive recursion. (shrink)

We investigate operators which take a set X to a set relatively computably enumerable in and above X by studying which such sets X can be so mapped into the Turing degree of K. We introduce notions of nontriviality for such operators, and use these to study which additional properties can be required of sets which can be completed to the jump by given operators of this kind.

We show the existence of a high d. c. e. degree d and a low2 c.e. degree a such that d is isolated by a.

The connected and acyclic components contained in a network are identified by the computation of its complexity, where complexity of a network refers to the total number of spanning trees present within. The article in hand deals with the enumeration of the complexity of various networks’ operations such as sum, product, difference K 2, n ⊖ K 2, and the conjunction of S n with K 2. All our computations have been concluded by implementation of the methods of linear (...) algebra and matrix theory. Our derivations will also be highlighted with the assistance of 3D plots at the end of this article. (shrink)

We introduce the computable FS-jump, an analog of the classical Friedman–Stanley jump in the context of equivalence relations on the natural numbers. We prove that the computable FS-jump is proper with respect to computable reducibility. We then study the effect of the computable FS-jump on computably enumerable equivalence relations (ceers).

We investigate the relationship of the degrees of splittings of a computably enumerable set and the degree of the set. We prove that there is a high computably enumerable set whose only proper splittings are low 2.

For a ring R, Hilbert’s Tenth Problem $HTP$ is the set of polynomial equations over R, in several variables, with solutions in R. We view $HTP$ as an enumeration operator, mapping each set W of prime numbers to $HTP$, which is naturally viewed as a set of polynomials in $\mathbb {Z}[X_1,X_2,\ldots ]$. It is known that for almost all W, the jump $W'$ does not $1$ -reduce to $HTP$. In contrast, we show that every Turing degree contains a set (...) W for which such a $1$ -reduction does hold: these W are said to be HTP-complete. Continuing, we derive additional results regarding the impossibility that a decision procedure for $W'$ from $HTP$ can succeed uniformly on a set of measure $1$, and regarding the consequences for the boundary sets of the $HTP$ operator in case $\mathbb {Z}$ has an existential definition in $\mathbb {Q}$. (shrink)

This study is an in-depth examination of the allure of Cartesianism. Its central focus is to uncover the grounds of Cartesianism in the will, and to show how such a grounding accounts for Descartes' immediate popularity and expansive influence. Cartesianism is generally taken to be a species of rationalism or foundationalism. However, it is essential to understanding Cartesianism to see that it has its foundations in an act of pure will. ;This rarely discussed aspect of the grounds of Descartes' method (...) determines every aspect of Cartesianism and is most accessible in the Discourse on Method. However, this is one of the only studies to examine The Rules for the Direction of the Mind from the perspective of Cartesian willfulness. Enumeration operates in The Rules in lieu of prudence and memory, making it appear that the method can be successful without a moment of judgment and concealing the shortcomings of the method which Descartes was not yet ready to admit as clearly as he would later in the Discourse. ;Vico was perhaps the first philosopher to appreciate the foundation of Cartesianism in the will. He was certainly the first to articulate the implications of this facet of Cartesianism in detail. The significance of the primacy of the will at the inception of Cartesianism is that a philosophical comportment to the world, oriented to an apprehension and articulation of the truth of the whole is sacrificed for the possibility of certainty and mastery. Vico defends the traditional project of philosophy against the Cartesian attempt to redefine the paradigm, and he does so in modern terms, in direct response to Descartes. ;Grounded in pure willfulness, Cartesian modernism engendered a myriad of constructivist projects which has polarized the history and field of philosophy. Those who pursue the Cartesian ideal without a sense of irony do so at the expense of a belief to the intelligibility of their experience, for their studies take place in abstract, imaginary, and hypothetical realms. Post-moderns, who have rejected Cartesian method, also abandon the possibility of the intelligibility of their experience for they are unable to fathom non-Cartesian forms of intelligibility. The only philosophical response to Cartesianism in the modern era is to understand it well enough to recognize its influence on our understanding, and try to transcend that influence. (shrink)

We prove an effective version of the Lopez-Escobar theorem for continuous domains. Let $Mod(\tau )$ be the set of countable structures with universe $\omega $ in vocabulary $\tau $ topologized by the Scott topology. We show that an invariant set $X\subseteq Mod(\tau )$ is $\Pi ^0_\alpha $ in the Borel hierarchy of this topology if and only if it is definable by a $\Pi ^p_\alpha $ -formula, a positive $\Pi ^0_\alpha $ formula in the infinitary logic $L_{\omega _1\omega }$. As (...) a corollary of this result we obtain a new pullback theorem for positive computable embeddings: Let $\mathcal {K}$ be positively computably embeddable in $\mathcal {K}'$ by $\Phi $, then for every $\Pi ^p_\alpha $ formula $\xi $ in the vocabulary of $\mathcal {K}'$ there is a $\Pi ^p_\alpha $ formula $\xi ^{*}$ in the vocabulary of $\mathcal {K}$ such that for all $\mathcal {A}\in \mathcal {K}$, $\mathcal {A}\models \xi ^{*}$ if and only if $\Phi (\mathcal {A})\models \xi $. We use this to obtain new results on the possibility of positive computable embeddings into the class of linear orderings. (shrink)

A proof is given that 0′ is definable in the structure of the degrees of unsolvability. This answers a long-standing question of Kleene and Post, and has a number of corollaries including the definability of the jump operator.

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable propositions, (...) and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic two-dimensional semantics. Chapter \textbf{3} provides an abstraction principle for epistemic (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between my hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between the interpretational and objective modalities and truthmakers thereof. This yields the first hyperintensional category theory in the literature. I invent a new mathematical trick in which first order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals while bypassing the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

Contents: Recursive Enumerability and the Jump Operator; On the Degrees Less Than 0'; A Simple Set Which Is Not Effectively Simple; The Recursively Enumerable Degrees Are Dense; Metarecursive Sets (with G Kreisel); Post's Problem, Admissible Ordinals and Regularity; On a Theorem of Lachlan and Marlin; A Minimal Hyperdegree (with R O Gandy); Measure-Theoretic Uniformity in Recursion Theory and Set Theory; Forcing with Perfect Closed Sets; Recursion in Objects of Finite Type; The a-Finite Injury Method (with S G Simpson); Remarks Against (...) Foundational Activity; Countable Admissible Ordinals and Hyperdegrees; The 1-Section of a Type n Object; The k-Section of a Type n Object; Post's Problem, Absoluteness and Recursion in Finite Types; Effective Bounds on Morley Rank; On the Number of Countable Models; Post's Problem in E-Recursion; The Limits of E-Recursive Enumerability; Effective Versus Proper Forcing. (shrink)

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable propositions, (...) and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter 2 argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic epistemic two-dimensional hyperintensional semantics. Chapter 3 provides an abstraction principle for epistemic (hyper-)intensions. Chapter 4 advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter 5 applies the fixed points of the modal μ-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal μ-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal μ-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter 6 advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter 7 provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters 2 and 4 are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. Chapters 8-12 provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter 8 examines the interaction between my hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of Ω-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter 9 examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter 10 examines the philosophical significance of hyperintensional Ω-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter 11 provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter 12 avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between the interpretational and objective modalities and truthmakers thereof. This yields the first hyperintensional category theory in the literature. I invent a new mathematical trick in which first order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals while bypassing the category of Set in category theory. Chapter 13 examines modal responses to the alethic paradoxes. Chapter 14 examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

It is shown the complete equivalence between the theory of continuous (enumeration) fuzzy closure operators and the theory of (effective) fuzzy deduction systems in Hilbert style. Moreover, it is proven that any truth-functional semantics whose connectives are interpreted in [0,1] by continuous functions is axiomatizable by a fuzzy deduction system (but not by an effective fuzzy deduction system, in general).

A theorem is derived that enables a systematic enumeration of all the linear superoperators ℒ (associated with a two-level quantum system) that generate, via the law of motion ℒρ= $\dot \rho$ , mappings ρ(0) → ρ(t) restricted to the domain of statistical operators. Such dynamical evolutions include the usual Hamiltonian motion as a special case, but they also encompass more general motions, which are noncyclic and feature a destination state ρ(t → ∞) that is in some cases independent (...) of ρ(0). (shrink)

We study effectively inseparable prelattices $\wedge, \vee$ are binary computable operations; ${ \le _L}$ is a computably enumerable preordering relation, with $0{ \le _L}x{ \le _L}1$ for every x; the equivalence relation ${ \equiv _L}$ originated by ${ \le _L}$ is a congruence on L such that the corresponding quotient structure is a nontrivial bounded lattice; the ${ \equiv _L}$ -equivalence classes of 0 and 1 form an effectively inseparable pair of sets). Solving a problem in we show, that if (...) L is an e.i. prelattice then ${ \le _L}$ is universal with respect to all c.e. preordering relations, i.e., for every c.e. preordering relation R there exists a computable function f reducing R to ${ \le _L}$, i.e., $xRy$ if and only if $f\left{ \le _L}f\left$, for all $x,y$. In fact ${ \le _L}$ is locally universal, i.e., for every pair $a{ < _L}b$ and every c.e. preordering relation R one can find a reducing function f from R to ${ \le _L}$ such that the range of f is contained in the interval $\left\{ {x:a{ \le _L}x{ \le _L}b} \right\}$. Also ${ \le _L}$ is uniformly dense, i.e., there exists a computable function f such that for every $a,b$ if $a{ < _L}b$ then $a{ < _L}f\left{ < _L}b$, and if $a{ \equiv _L}a\prime$ and $b{ \equiv _L}b\prime$ then $f\left{ \equiv _L}f\left$. Some consequences and applications of these results are discussed: in particular for $n \ge 1$ the c.e. preordering relation on ${{\rm{\Sigma }}_n}$ sentences yielded by the relation of provable implication of any c.e. consistent extension of Robinson’s system R or Q is locally universal and uniformly dense; and the c.e. preordering relation yielded by provable implication of any c.e. consistent extension of Heyting Arithmetic is locally universal and uniformly dense. (shrink)

As a natural example of a 1-random real, Chaitin proposed the halting probability Ω of a universal prefix-free machine. We can relativize this example by considering a universal prefix-free oracle machine U. Let [Formula: see text] be the halting probability of UA; this gives a natural uniform way of producing an A-random real for every A ∈ 2ω. It is this operator which is our primary object of study. We can draw an analogy between the jump operator from computability theory (...) and this Omega operator. But unlike the jump, which is invariant under the choice of an effective enumeration of the partial computable functions, [Formula: see text] can be vastly different for different choices of U. Even for a fixed U, there are oracles A =* B such that [Formula: see text] and [Formula: see text] are 1-random relative to each other. We prove this and many other interesting properties of Omega operators. We investigate these operators from the perspective of analysis, computability theory, and of course, algorithmic randomness. (shrink)

Let $V_\propto$ be a fixed, fully effective, infinite dimensional vector space. Let $\mathscr{L}(V_\propto)$ be the lattice consisting of the recursively enumerable (r.e.) subspaces of $V_\propto$ , under the operations of intersection and weak sum (see § 1 for precise definitions). In this article we examine the algebraic properties of $\mathscr{L}(V_\propto)$ . Early research on recursively enumerable algebraic structures was done by Rabin [14], Frolich and Shepherdson [5], Dekker [3], Hamilton [7], and Guhl [6]. Our results are based upon the more (...) recent work concerning vector spaces of Metakides and Nerode [12], Crossley and Nerode [2], Remmel [15], [16], and Kalantari [8]. In the main theorem below, we extend a result of Lachlan from the lattice E of r.e. sets to $\mathscr{L}(V_\propto)$ . We define hyperhypersimple vector spaces, discuss some of their properties and show if $A, B \in \mathscr{L}(V_\propto)$ , and A is a hyperhypersimple subspace of B then there is a recursive space C such that A + C = B. It will be proven that if $V \in \mathscr{L}(V_\propto)$ and the lattice of superspaces of V is a complemented modular lattice then V is hyperhypersimple. The final section contains a summary of related results concerning maximality and simplicity. (shrink)

In 1950, B.A. Trakhtenbrot showed that the set of first-order tautologies associated to finite models is not recursively enumerable. In 1999, P. Hájek generalized this result to the first-order versions of Łukasiewicz, Gödel and Product logics, w.r.t. their standard algebras. In this paper we extend the analysis to the first-order versions of axiomatic extensions of MTL. Our main result is the following. Let \ be a class of MTL-chains. Then the set of all first-order tautologies associated to the finite models (...) over chains in \, fTAUT\, is \ -hard. Let TAUT\ be the set of propositional tautologies of \. If TAUT\ is decidable, we have that fTAUT\ is in \. We have similar results also if we expand the language with the Δ operator. (shrink)

We investigate systems of ordinary differential equations with a parameter. We show that under suitable assumptions on the systems the solutions are computable in the sense of recursive analysis. As an application we give a complete characterization of the recursively enumerable sets using Fourier coefficients of recursive analytic functions that are generated by differential equations and elementary operations.

We investigate computational properties of propositional logics for dynamical systems. First, we consider logics for dynamic topological systems (W.f), fi, where W is a topological space and f a homeomorphism on W. The logics come with ‘modal’ operators interpreted by the topological closure and interior, and temporal operators interpreted along the orbits {w, f(w), f2 (w), ˙˙˙} of points w ε W. We show that for various classes of topological spaces the resulting logics are not recursively enumerable (and (...) so not recursively axiomatisable). This gives a ‘negative’ solution to a conjecture of Kremer and Mints. Second, we consider logics for dynamical systems (W, f), where W is a metric space and f and isometric function. The operators for topological interior/closure are replaced by distance operators of the form ‘everywhere/somewhere in the ball of radius a, ‘for a ε Q +. In contrast to the topological case, the resulting logic turns out to be decidable, but not in time bounded by any elementary function. (shrink)

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable propositions, (...) and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic two-dimensional semantics. Chapter \textbf{3} provides an abstraction principle for epistemic (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between my hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between the interpretational and objective modalities and truthmakers thereof. This yields the first hyperintensional category theory in the literature. I invent a new mathematical trick in which first order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals while bypassing the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

Charles-Louis de Secondat, baron de La Brède et de Montesquieu, mentions Niccolò Machiavelli by name in his extant works just a handful of times. That, however, he read him carefully and thoroughly time and again there can be no doubt, and it is also clear that he couches his argument both in his Considerations on the Causes of the Greatness of the Romans and their Decline and in his Spirit of Laws as an appropriation and critique of the work of (...) the predecessor whom he termed ‘this great man’. In this paper I explore the manner in which the Frenchman redeployed the arguments advanced by the Florentine for the purpose of refuting the latter's conclusions. ☆ In citing Machiavelli, I employ N. Machiavelli, Tutte le opere, ed. M. Martelli (Florence, 1971), and I refer to the divisions within each work provided by the author. To make my references to Machiavelli's Discorsi sopra la prima deca di Tito Livio more precise, I sometimes also employ the paragraph enumeration added by the editors of N. Machiavelli, Discourses on Livy, tr. H. C. Mansfield and N. Tarcov (Chicago, 1996). In citing Montesquieu, wherever possible, I have employed the splendid new critical edition being produced by the Société Montesquieu: Charles-Louis de Secondat, baron de La Brède et de Montesquieu, Œuvres complètes de Montesquieu, ed. J. Ehrard, C. Volpilhac-Auger, et al., 22 vols. (Oxford, 1998-), which I cite as VF. I cite Montesquieu, Considérations sur les causes de la grandeur des Romains et de leur décadence (1734), ed. F. Weil and C. Courtney, as CR by chapter and, where appropriate, line from VF, ii, 89–285. I cite Montesquieu, De l’Esprit des lois (1757), as EL by part, book, chapter, and, where appropriate, page from Montesquieu, Œuvres complètes de Montesquieu, ed. R. Caillois, 2 vols. (Paris, 1949–51), ii, 225–995, and Charles-Louis de Secondat, baron de La Brède et de Montesquieu, Mes pensées as MP by number from Montesquieu, Œuvres complètes de Montesquieu, ed. A. Masson, 3 vols. (Paris, 1950–5), ii, 1–677. (shrink)

The common view of the transition between subitizing and numerosity estimation regimes is that there is a hard bound on the subitizing range, and beyond this range, people estimate. However, this view does not adequately address the behavioral signatures of enumeration under conditions of attentional load or in the immediate post-subitizing range. The possibility that there might exist a numerosity range where both processes of subitizing and estimation operate in conjunction has so far been ignored. Here, we investigate this (...) new proposal, that people strategically combine the processes of subitizing and estimation to maximize accuracy and precision, given time or attentional constraints. We present a process-level account of how subitizing and estimation can be combined through strategic deployment of attention to maximize the precision of perceived numerosity given time constraints. We then describe a computational model of this account and apply it in two experimental simulations to demonstrate how it can explain key findings in prior enumeration research. While recent modeling work has argued that the behavioral signatures of enumeration can best be explained through a single numerosity system with a single form of representation, we argue that our model demonstrates how the traditional two-systems view of numerical representation accounts for behavioral data through coordination with a unified attentional mechanism, rather than a unified representation. (shrink)

The modern theory of computability is based on the works of Church, Markov and Turing who, starting from quite different models of computation, arrived at the same class of computable functions. The purpose of this paper is the show how the main results of the Church-Markov-Turing theory of computable functions may quickly be derived and understood without recourse to the largely irrelevant theories of recursive functions, Markov algorithms, or Turing machines. We do this by ignoring the problem of what constitutes (...) a computable function and concentrating on the central feature of the Church-Markov-Turing theory: that the set of computable partial functions can be effectively enumerated. In this manner we are led directly to the heart of the theory of computability without having to fuss about what a computable function is.The spirit of this approach is similar to that of [RGRS]. A major difference is that we operate in the context of constructive mathematics in the sense of Bishop [BSH1], so all functions are computable by definition, and the phrase “you can find” implies “by a finite calculation.” In particular ifPis some property, then the statement “for eachmthere isnsuch thatP” means that we can construct a functionθsuch thatP)for allm. Church's thesis has a different flavor in an environment like this where the notion of a computable function is primitive.One point of such a treatment of Church's thesis is to make available to Bishopstyle constructivists the Markovian counterexamples of Russian constructivism and recursive function theory. The lack of serious candidates for computable functions other than recursive functions makes it quite implausible that a Bishopstyle constructivist could refute Church's thesis, or any consequence of Church's thesis. Hence counterexamples such as Specker's bounded increasing sequence of rational numbers that is eventually bounded away from any given real number [SPEC] may be used, as Brouwerian counterexamples are, as evidence of the unprovability of certain assertions. (shrink)

The recursion-theoretic study of mathematical structures other thanωis now a field of much activity. Analysis and algebra are two such structures which have been studied for their effective contents. Studies in analysis began with the work of Specker on nonconstructive proofs in analysis [16] and in Lacombe's inspiring notes on relevant notions of recursive analysis [8]. Studies in algebra originated in the work of Frolich and Shepherdson on effective extensions of explicit fields [1] and in Rabin's study of computable fields (...) [15]. Equipped with the richness of modern techniques in recursion theory, Metakides and Nerode [11]–[13] began investigating the effective content of vector spaces and fields; these studies have been extended by Kalantari, Remmel, Retzlaff, Shore and others.Kalantari and Retzlaff [5] began a foundational inquiry into effectiveness in topological spaces. They consider a topological spaceXwith a countable basis ⊿ for the topology. The space isfully effective, that is, the basis elements are coded intoωand the operation of intersection of basis elements and the relation of inclusion among them are both computable. Similar to, the lattice of recursively enumerable subsets ofω, the collection of r.e. open subsets ofXforms a latticeℒunder the usual operations of union and intersection. (shrink)

It has been argued that 'brain bisection' data leads us to abandon our traditional conception of personal identity. Nagel has remarked: The ultimate account of the unity of what we call a single mind consists of an enumeration of the types of functional integration that typify it. We know that these can be eroded in different ways and to different degrees. The belief that even in their complete version they can be explained by the presence of a numerically single (...) subject is an i1lusion.l Parfit has adopted a similar position, contending that patients with 'split brains' become two separate 'streams of consciousness' and thus that our normal sense of personal identity, or at least 'what matters' about personal identity is constituted by psychological relations between connected conscious experiences2 It is claimed that in 'split brain' patients certain of the relations are disrupted and that we thus see clearly that the nature of the unity that is normally present does not reside in a single subject with a given identity, but in the connectedness and continuity that normally obtains. Parfit draws on two sources of support for these contentions: the first is the actual events that transpire after a human being is submitted to the operation of sectioning the corpus callosum (or 'brain bisection'), and the second is the imaginative consideration of various scenarios involving graded mental and physical discontinuity, and the 'fission' and 'fusion' of persons. I shall do little more than argue that the actual data will not sustain the interpretation put on them. (shrink)

We propose and investigate a uniform modal logic framework for reasoning about topology and relative distance in metric and more general distance spaces, thus enabling the comparison and combination of logics from distinct research traditions such as Tarski’s for topological closure and interior, conditional logics, and logics of comparative similarity. This framework is obtained by decomposing the underlying modal-like operators into first-order quantifier patterns. We then show that quite a powerful and natural fragment of the resulting first-order logic can (...) be captured by one binary operator comparing distances between sets and one unary operator distinguishing between realised and limit distances . Due to its greater expressive power, this logic turns out to behave quite differently from both and conditional logics. We provide finite axiomatisations and ExpTime-completeness proofs for the logics of various classes of distance spaces, in particular metric spaces. But we also show that the logic of the real line is not recursively enumerable. This result is proved by an encoding of Diophantine equations. (shrink)

A type-structure of partial effective functionals over the natural numbers, based on a canonical enumeration of the partial recursive functions, is developed. These partial functionals, defined by a direct elementary technique, turn out to be the computable elements of the hereditary continuous partial objects; moreover, there is a commutative system of enumerations of any given type by any type below (relative numberings). By this and by results in [1] and [2], the Kleene-Kreisel countable functionals and the hereditary effective operations (...) (HEO) are easily characterized. (shrink)

Computability theory concerns information with a causal–typically algorithmic–structure. As such, it provides a schematic analysis of many naturally occurring situations. Emil Post was the first to focus on the close relationship between information, coded as real numbers, and its algorithmic infrastructure. Having characterised the close connection between the quantifier type of a real and the Turing jump operation, he looked for more subtle ways in which information entails a particular causal context. Specifically, he wanted to find simple relations on reals (...) which produced richness of local computability-theoretic structure. To this extent, he was not just interested in causal structure as an abstraction, but in the way in which this structure emerges in natural contexts. Post’s programme was the genesis of a more far reaching research project. In this article we will firstly review the history of Post’s programme, and look at two interesting developments of Post’s approach. The first of these developments concerns the extension of the core programme, initially restricted to the Turing structure of the computably enumerable sets of natural numbers, to the Ershov hierarchy of sets. The second looks at how new types of information coming from the recent growth of research into randomness, and the revealing of unexpected new computability-theoretic infrastructure. We will conclude by viewing Post’s programme from a more general perspective. We will look at how algorithmic structure does not just emerge mathematically from information, but how that emergent structure can model the emergence of very basic aspects of the real world. (shrink)