The biblical sentence “God is Spirit” (John 4:24) occasioned the development of the Christian doctrine about God as Spirit. But since patristic times “spirit” was interpreted in the sense of Nus, which rather means “intellect.” The biblical concept of spirit (pneuma), however, has its root meaning in referring to “air in movement,” as in breath or storm. The similar concept of pneuma in Stoic philosophy has become the “immediate precursor” (Max Jammer) of the field concept in modern physics, so that (...) the conclusion is suggested that God is spirit as something like a field of force rather than as intellect. This essay argues for such a conception by relating the divine eternity and immensity to the concepts of space and time, the basic requirements of any physical field. God's eternity and immensity are interpreted in terms of undivided infinite space (and time) which is presupposed in all concepts of parts of space or time (or space‐time), therefore in all mathematical and physical measurement. (shrink)

We define a class of productive σ-ideals of subsets of the Cantor space 2 ω and observe that both σ-ideals of meagre sets and of null sets are in this class. From every productive σ-ideal I we produce a σ-ideal I κ , of subsets of the generalized Cantor space 2 κ . In particular, starting from meagre sets and null sets in 2 ω we obtain meagre sets and null sets in 2 κ , respectively. (...) Then we investigate additivity, covering number, uniformity and cofinality of I κ . For example, we show that non(I = non(I ω 1 ) = non(I ω 2 ). Our results generalizes those from [5]. (shrink)

Let $\mathcal{L}$ be a propositional language with standard Boolean connectives plus two modalities: an S4-ish topological modality $\square$ and a temporal modality $\bigcirc$ , understood as ‘next’. We extend the topological semantic for S4 to a semantics for the language $\mathcal{L}$ by interpreting $\mathcal{L}$ in dynamic topological systems, i.e. ordered pairs $\langle X, f\rangle$ , where X is a topological space and f is a continuous function on X. Artemov, Davoren and Nerode have axiomatized a logic S4C, and have (...) shown that S4C is sound and complete for this semantics. Zhang and Mints have shown that S4C is complete relative to a particular topological space, Cantor space. The current paper produces an alternate proof of the Zhang-Mints result. (shrink)

In this paper, we are interested in parallels to the classical notions of special subsets in defined in the generalized Cantor and Baire spaces (2κ and ). We consider generalizations of the well‐known classes of special subsets, like Lusin sets, strongly null sets, concentrated sets, perfectly meagre sets, σ‐sets, γ‐sets, sets with the Menger, the Rothberger, or the Hurewicz property, but also of some less‐know classes like X‐small sets, meagre additive sets, Ramsey null sets, Marczewski, Silver, Miller, and Laver‐null (...) sets. We notice that many classical theorems regarding these classes can be relatively easy generalized to higher cardinals although sometimes with some additional assumptions. This paper serves as a catalogue of such results along with some other generalizations and open problems. (shrink)

This paper has two parts. In the first, I try to show that Russell's arguments against the thesis of Kant's first antinomy are unsatisfactory; in the second, I argue that the Universe, if transcendentally real, must be finite in both space and time.

It is well known that infinite minimal sets for continuous functions on the interval are Cantor sets; that is, compact zero dimensional metrizable sets without isolated points. On the other hand, it was proved in Alcaraz and Sanchis (Bifurcat Chaos 13:1665–1671, 2003) that infinite minimal sets for continuous functions on connected linearly ordered spaces enjoy the same properties as Cantor sets except that they can fail to be metrizable. However, no examples of such subsets have been known. In (...) this note we construct, in ZFC, ${2^{\mathfrak{c}}}$ non-metrizable infinite pairwise non-homeomorphic minimal sets on compact connected linearly ordered spaces. (shrink)

We develop the general theory of topometric spaces, i.e., topological spaces equipped with a well-behaved lower semi-continuous metric. Spaces of global and local types in continuous logic are the motivating examples for the study of such spaces. In particular, we develop Cantor-Bendixson analysis of topometric spaces, which can serve as a basis for the study of local stability (extending the ad hoc development in Ben Yaacov I and Usvyatsov A, Continuous first order logic and local stability. Trans Am Math (...) Soc, in press), as well as of global ${\aleph_0}$ -stability. We conclude with a study of perturbation systems (see Ben Yaacov I, On perturbations of continuous structures, submitted) in the formalism of topometric spaces. In particular, we show how the abstract development applies to ${\aleph_0}$ -stability up to perturbation. (shrink)

The concept of a Mannigfaltigkeit in Husserl has been given various interpretations, due to its shifting role in his works. Many authors have been misled by this term, placing it in the context of Husserl’s early period in Halle, while writing the Philosophy of Arithmetic, as a friend and colleague of Georg Cantor.Yet at the time, Husserl distanced himself explicitly from Cantor’s definition and rather took Bernhard Riemann as example, having studied and lectured extensively on Riemann’s theories of (...) space. Husserl’s Mannigfaltigkeitslehre would then not be a Cantorian set-theory, but come rather closer to topology. Then, in the Prolegomena, Husserl introduces the idea of a pure Mannigfaltigkeitslehre as a meta-theoretical enterprise which studies the relations among theories, e.g. how to derive or found one upon another. When Husserl announces that in fact the best example of such a pure theory of manifolds is what is actually practiced in mathematics, this sounds slightly misleading. The pure theory of theories cannot simply be the mathematics underlying topology, but should rather be considered as a mathesis universalis. Indeed, while this might not have been fully clear yet in 1900/1901, Husserl will explicitly tie together the notions of pure theory of manifolds and mathesis universalis. The mathesis universalis in this sense is formal, a priori and analytic, as theory of theory in general. It is an analysis of the highest categories of meaning and their correlative categories of objects. In my paper I try to understand the development of the notion of Mannigfaltigkeit in Husserl’s thought from its mathematical beginnings to its later central philosophical role, taking into account the mathematical background and context of Husserl’s own development. (shrink)

This paper continues the program connecting reverse mathematics and computable analysis via the framework of Weihrauch reducibility. In particular, we consider problems related to perfect subsets of Polish spaces, studying the perfect set theorem, the Cantor–Bendixson theorem, and various problems arising from them. In the framework of reverse mathematics, these theorems are equivalent, respectively, to $\mathsf {ATR}_0$ and $\boldsymbol {\Pi }^1_1{-}\mathsf{CA}_0$, the two strongest subsystems of second order arithmetic among the so-called big five. As far as we know, this (...) is the first systematic study of problems at the level of $\boldsymbol {\Pi }^1_1{-}\mathsf{CA}_0$ in the Weihrauch lattice. We show that the strength of some of the problems we study depends on the topological properties of the Polish space under consideration, while others have the same strength once the space is rich enough. (shrink)

In the topological semantics for modal logic, S4 is well-known to be complete for the rational line, for the real line, and for Cantor space: these are special cases of S4’s completeness for any dense-in-itself metric space. The construction used to prove completeness can be slightly amended to show that S4 is not only complete, but also strongly complete, for the rational line. But no similarly easy amendment is available for the real line or for Cantor (...) space and the question of strong completeness for these spaces has remained open, together with the more general question of strong completeness for any dense-in-itself metric space. In this paper, we prove that S4 is strongly complete for any dense-in-itself metric space. (shrink)

Computability theoretic aspects of Polish metric spaces are studied by adapting notions and methods of computable structure theory. In this dissertation, we mainly investigate index sets and classification problems for computably presentable Polish metric spaces. We find the complexity of a number of index sets, isomorphism problems, and embedding problems for computably presentable metric spaces. We also provide several computable structure theory results related to some classical Polish metric spaces such as the Urysohn space $\mathbb {U}$, the Cantor (...) space $2^{\mathbb {N}}$, the Baire space $\mathbb {N}^{\mathbb {N}}$, and spaces of continuous functions.Abstract prepared by Teerawat Thewmorakot.E-mail: teerawat.thew@hotmail.com. (shrink)

We establish some results on the Borel and difference hierarchies in φ-spaces. Such spaces are the topological counterpart of the algebraic directed-complete partial orderings. E.g., we prove analogs of the Hausdorff Theorem relating the difference and Borel hierarchies and of the Lavrentyev Theorem on the non-collapse of the difference hierarchy. Some of our results generalize results of A. Tang for the space Pω. We also sketch some older applications of these hierarchies and present a new application to the question (...) of characterizing the ω-ary Boolean operations generating a given level of the Wadge hierarchy from the open sets. (shrink)

We investigate the Ramsey theory of continuous graph-structures on complete, separable metric spaces and apply the results to the problem of covering a plane by functions. Let the homogeneity number[Formula: see text] of a pair-coloring c:[X]2→2 be the number of c-homogeneous subsets of X needed to cover X. We isolate two continuous pair-colorings on the Cantor space 2ω, c min and c max, which satisfy [Formula: see text] and prove: Theorem. For every Polish space X and every (...) continuous pair-coloringc:[X]2→2with[Formula: see text], [Formula: see text] There is a model of set theory in which[Formula: see text]and[Formula: see text]. The consistency of [Formula: see text] and of [Formula: see text] follows from [20]. We prove that [Formula: see text] is equal to the covering number of 2 by graphs of Lipschitz functions and their reflections on the diagonal. An iteration of an optimal forcing notion associated to c min gives: Theorem. There is a model of set theory in which ℝ2 is coverable byℵ1graphs and reflections of graphs of continuous real functions; ℝ2 is not coverable byℵ1graphs and reflections of graphs of Lipschitz real functions. Figure 1.1 in the introduction summarizes the ZFC results in Part I of the paper. The independence results in Part II show that any two rows in Fig. 1.1 can be separated if one excludes [Formula: see text] from row. (shrink)

Could space consist entirely of extended regions, without any regions shaped like points, lines, or surfaces? Peter Forrest and Frank Arntzenius have independently raised a paradox of size for space like this, drawing on a construction of Cantor’s. I present a new version of this argument and explore possible lines of response.

Dynamic Topological Logic ( $\mathcal{DTL}$ ) is a modal framework for reasoning about dynamical systems, that is, pairs 〈X, f〉 where X is a topological space and f: X → X a continuous function. In this paper we consider the case where X is a metric space. We first show that any formula which can be satisfied on an arbitrary dynamic topological system can be satisfied on one based on a metric space; in fact, this space (...) can be taken to be countable and have no isolated points. Since any metric space with these properties is homeomorphic to the set of rational numbers, it follows that any satisfiable formula can be satisfied on a system based on $\mathbb{Q}$ . We then show that the situation changes when considering complete metric spaces, by exhibiting a formula which is not valid in general but is valid on the class of systems based on a complete metric space. While we do not attempt to give a full characterization of the set of valid formulas on this class we do give a relative completeness result; any formula which is satisfiable on a dynamical system based on a complete metric space is also satisfied on one based on the Cantor space. (shrink)

Unlike Martin‐Löf randomness and Schnorr randomness, computable randomness has not been defined, except for a few ad hoc cases, outside of Cantor space. This paper offers such a definition (actually, several equivalent definitions), and further, provides a general method for abstracting “bit‐wise” definitions of randomness from Cantor space to arbitrary computable probability spaces. This same method is also applied to give machine characterizations of computable and Schnorr randomness for computable probability spaces, extending the previously known results. (...) The paper contains a new type of randomness—endomorphism randomness—which the author hopes will shed light on the open question of whether Kolmogorov‐Loveland randomness is equivalent to Martin‐Löf randomness. The last section contains ideas for future research. (shrink)

We study the Medvedev degrees of mass problems with distinguished topological properties, such as denseness, closedness, or discreteness. We investigate the sublattices generated by these degrees; the prime ideal generated by the dense degrees and its complement, a prime filter; the filter generated by the nonzero closed degrees and the filter generated by the nonzero discrete degrees. We give a complete picture of the relationships of inclusion holding between these sublattices, these filters, and this ideal. We show that the sublattice (...) of the closed Medvedev degrees is not a Brouwer algebra. We investigate the dense degrees of mass problems that are closed under Turing equivalence, and we prove that the dense degrees form an automorphism base for the Medvedev lattice. The results hold for both the Medvedev lattice on the Baire space and the Medvedev lattice on the Cantor space. (shrink)

INSPIRED by a dynamist Naturphilosophie and looking for a mathematics of the natura naturans, the founders of modern mathematics in Germany made some lasting contributions in the attempt to go beyond perceptible space. Hermann Grassmann’s extension theory, Johann Benedict Listing’s topology, Bernhard Riemann’s non-Euclidean manifold theory, Carl Gustav Jacob Jacobi’s approach to non-mechanistic theory and last but not least Georg Cantor’s transfinite set theory were all influenced by the tradition of Naturphilosophie. One central motivation for the new mathematics (...) was to decipher the so-called “inner side of nature” which was thought to be invisible but nonetheless physical. As a result, mathematics was understood not only in terms of quantity but of morphogenetic quality and as a tool to recognize the intrinsic possibilities of structuring and organizing nature. This essay focuses on the philosophical viewpoints of Christian Samuel Weiss, Justus and Hermann Grassmann, Bernhard Riemann and their reception of Schelling’s philosophy. Riemann deconstructed empiricist theories of space so as to glean the fundamental constitution of space. One result was that a given, fixed and static metric is not necessary for the understanding of space. The underlying topological substratum has no specific metric but is capable of exhibiting any possible metric. Riemann’s claims are compared with Schelling’s theory of space in his “Darstellung des Naturprocesses”. My thesis is that Schelling’s philosophy of the emergence of space can be read in Riemannian terms and that it is an interesting model in the context of modern questions. (shrink)

We work in the Baire space $$\mathbb {Z}^\omega$$ equipped with the coordinate-wise addition $$+$$. Consider a $$\sigma -$$ ideal $$\mathcal {I}$$ and a family $$\mathbb {T}$$ of some kind of perfect trees. We are interested in results of the form: for every $$A\in \mathcal {I}$$ and a tree $$T\in \mathbb {T}$$ there exists $$T'\in \mathbb {T}, T'\subseteq T$$ such that $$A+\underbrace{[T']+[T']+\dots +[T']}_{\text {n--times}}\in \mathcal {I}$$ for each $$n\in \omega$$. Explored tree types include perfect trees, uniformly perfect trees, (...) Miller trees, Laver trees and $$\omega -$$ Silver trees. The latter kind of trees is an analogue of Silver trees from the Cantor space. Besides the standard $$\sigma$$ -ideal $$\mathcal {M}$$ of meager sets, we also analyze $$\mathcal {M}_-$$ and fake null sets $$\mathcal {N}$$. The latter two are born out of the characterizations of their respective analogues in the Cantor space. The key ingredient in proofs were combinatorial characterizations of these ideals in the Baire space. (shrink)

We investigate the notion of strong measure zero sets in the context of the higher Cantor space $2^\kappa $ for $\kappa $ at least inaccessible. Using an iteration of perfect tree forcings, we give two proofs of the relative consistency of $$\begin{align*}|2^\kappa| = \kappa^{++} + \forall X \subseteq 2^\kappa:\ X \textrm{ is strong measure zero if and only if } |X| \leq \kappa^+. \end{align*}$$ Furthermore, we also investigate the stronger notion of stationary strong measure zero and show that (...) the equivalence of the two notions is undecidable in ZFC. (shrink)

We investigate computability theoretic and topological properties of spaces of orders on computable orderable groups. A left order on a group G is a linear order of the domain of G, which is left-invariant under the group operation. Right orders and bi-orders are defined similarly. In particular, we study groups for which the spaces of left orders are homeomorphic to the Cantor set, and their Turing degree spectra contain certain upper cones of degrees. Our approach unifies and extends Sikora’s (...) [28] investigation of orders on groups in topology and Solomon’s [31] investigation of these orders in computable algebra. Furthermore, we establish that a computable free group Fn of rank n>1 has a bi-order in every Turing degree. (shrink)

We discuss the existence of universal spaces (either in the sense of embeddings or continuous images) for some classes of scattered Eberlein compacta. Given a cardinal κ, we consider the class Sκ of all scattered Eberlein compact spaces K of weight ≤ κ and such that the second Cantor-Bendixson derivative of K is a singleton. We prove that if κ is an uncountable cardinal such that κ = 2<κ, then there exists a space X in Sκ such that (...) every member of Sκ is homeomorphic to a retract of X. We show that it is consistent that there does not exist a universal space (either by embeddings or by mappings onto) in Sω₁. Assuming that ∂ = ω₁, we prove that there exists a space X ∈ Sω₁, which is universal in the sense of embeddings. We also show that it is consistent that there exists a space X ∈ Sω₁, universal in the sense of embeddings, but Sω₁ does not contain an universal element in the sense of mappings onto. (shrink)

In this paper we show that some standard topological constructions may be fruitfully used in the theory of closure spaces (see [5], [4]). These possibilities are exemplified by the classical theorem on the universality of the Alexandroff's cube for T 0-closure spaces. It turns out that the closure space of all filters in the lattice of all subsets forms a generalized Alexandroff's cube that is universal for T 0-closure spaces. By this theorem we obtain the following characterization of the (...) consequence operator of the classical logic: If is a countable set and C: P() P() is a closure operator on X, then C satisfies the compactness theorem iff the closure space ,C is homeomorphically embeddable in the closure space of the consequence operator of the classical logic.We also prove that for every closure space X with a countable base such that the cardinality of X is not greater than 2 there exists a subset X of irrationals and a subset X of the Cantor's set such that X is both a continuous image of X and a continuous image of X. (shrink)

본 연구는 연속체로서 공간의 형이상학적 기초를 논의한다. 공간의 특징인 연장성(extension)과 가분성(divisibility)에 따라 세 가지 공간 구성 후보가 있다: 포인트, 겅크, 청크. 포인트는 연장되지 않고 나뉘지 않는 것이다. 겅크는 그 반대로 연장성을 가지며 무한하게 나누어진다. 청크는 더 이상 나누어지지 않으면서도 연장되는 특징을 갖는다. 이 중에서 논리적으로 일관되면서 이론적으로 부담이 적은 양 극단의 두 개념, 포인트와 겅크를 분석한다. 이를 분석하는 도구로 공간 구조 분석의 기초인 위상구조(topology), 미레올로지(mereology), 측도론(measure theory)의 개념들을 사용한다.BR 고대 그리스부터 포인트는 영역 크기를 결정할 수 없게 만든다고 비판 받았다. 포인트 (...) 기반 위상구조 때문일 때도 있고, 포인트 기반 측도론 때문일 때도 있다. 본고에서는 그 중에서 두 퍼즐, 접촉 퍼즐과 나누기 퍼즐을 다룬다. 크기에 대한 문제는 포인트 관점에만 발생하지 않는다. 본고에서는 칸토어 구간(Cantor Interval)이라는 무한히 나뉠 수 있는 겅크 구조에서 발생하는 측도론 문제를 분석한다.BR 이 퍼즐들을 해소하기 위해 두 가지 방식의 전략을 논의한다. 하나는 토폴로지 개념인 경계 개념을 수정하는 방법이다. 다른 하나는 측도론의 원리인 가법성(additivity) 개념을 수정하는 방법이다. 본고는 전자의 문제점을 지적하고 후자의 접근법에서 기존의 공간 직관에 새로운 함축을 찾는다. 특히 미레올로지 구성과 일치하는 가법성과 집합 크기에 대응하는 가법성 구분을 공간 퍼즐들의 교훈으로 발견한다. (shrink)

A constructive definition of the continuum based on formal topology is given and its basic properties studied. A natural notion of Cauchy sequence is introduced and Cauchy completeness is proved. Other results include elementary proofs of the Baire and Cantor theorems. From a classical standpoint, formal reals are seen to be equivalent to the usual reals. Lastly, the relation of real numbers as a formal space to other approaches to constructive real numbers is determined.

We prove strong completeness results for some modal logics with the universal modality, with respect to their topological semantics over 0-dimensional dense-in-themselves metric spaces. We also use failure of compactness to show that, for some languages and spaces, no standard modal deductive system is strongly complete.

The present paper proposes a generalisation of the notion of disjunctive sequence, that is, of an infinite sequence of letters having each finite sequence as a subword. Our aim is to give a reasonable notion of disjunctiveness relative to a given set of sequences F. We show that a definition like “every subword which occurs at infinitely many different positions in sequences in F has to occur infinitely often in the sequence” fulfils properties similar to the original unrelativised notion of (...) disjunctiveness. Finally, we investigate our concept of generalised disjunctiveness in spaces of Cantor expansions of reals. (shrink)

We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space (CSM) without isolated points, X, in such a way that every sequentially continuous function from Cantor space to ℤ extends to a sequentially continuous function from X to ℝ. The second asserts an analogous property for Baire space relative to any inhabited locally non‐compact (...) CSM. Both results rely on having careful constructive formulations of the concepts involved.As a first application, we derive new relationships between “continuity principles” asserting that all functions between specified metric spaces are pointwise continuous. In particular, we give conditions that imply the failure of the continuity principle “all functions from X to ℝ are continuous”, when X is an inhabited CSM without isolated points, and when X is an inhabited locally non‐compact CSM. One situation in which the latter case applies is in models based on “domain realizability”, in which the failure of the continuity principle for any inhabited locally non‐compact CSM, X, generalizes a result previously obtained by Escardó and Streicher in the special case X = C[0, 1].As a second application, we show that, when the notion of inhabited complete separable metric space without isolated points is interpreted in a recursion‐theoretic setting, then, for any such space X, there exists a Banach‐Mazur computable function from X to the computable real numbers that is not Markov computable. This generalizes a result obtained by Hertling in the special case that X is the space of computable real numbers. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

We prove two embedding and extension theorems in the context of the constructive theory of metric spaces. The first states that Cantor space embeds in any inhabited complete separable metric space (CSM) without isolated points, X, in such a way that every sequentially continuous function from Cantor space to ℤ extends to a sequentially continuous function from X to ℝ. The second asserts an analogous property for Baire space relative to any inhabited locally non‐compact (...) CSM. Both results rely on having careful constructive formulations of the concepts involved.As a first application, we derive new relationships between “continuity principles” asserting that all functions between specified metric spaces are pointwise continuous. In particular, we give conditions that imply the failure of the continuity principle “all functions from X to ℝ are continuous”, when X is an inhabited CSM without isolated points, and when X is an inhabited locally non‐compact CSM. One situation in which the latter case applies is in models based on “domain realizability”, in which the failure of the continuity principle for any inhabited locally non‐compact CSM, X, generalizes a result previously obtained by Escardó and Streicher in the special case X = C[0, 1].As a second application, we show that, when the notion of inhabited complete separable metric space without isolated points is interpreted in a recursion‐theoretic setting, then, for any such space X, there exists a Banach‐Mazur computable function from X to the computable real numbers that is not Markov computable. This generalizes a result obtained by Hertling in the special case that X is the space of computable real numbers. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

Up to equivalence, a substitution in propositional logic is an endomorphism of its free algebra. On the dual space, this results in a continuous function, and whenever the space carries a natural measure one may ask about the stochastic properties of the action. In classical logic there is a strong dichotomy: while over finitely many propositional variables everything is trivial, the study of the continuous transformations of the Cantor space is the subject of an extensive literature, (...) and is far from being a completed task. In many-valued logic this dichotomy disappears: already in the finite-variable case many interesting phenomena occur, and the present paper aims at displaying some of these. (shrink)

We consider finite partitions of the closure F¯ of an ωα-uniform barrier F. For each partition, we get a homogeneous set having both the same combinatorial and topological structure as F¯, seen as a subspace of the Cantor space 2N.

We begin with the history of the discovery of computability in the 1930’s, the roles of Gödel, Church, and Turing, and the formalisms of recursive functions and Turing automatic machines . To whom did Gödel credit the definition of a computable function? We present Turing’s notion [1939, §4] of an oracle machine and Post’s development of it in [1944, §11], [1948], and finally Kleene-Post [1954] into its present form. A number of topics arose from Turing functionals including continuous functionals on (...) Cantor space and online computations. Almost all the results in theoretical computability use relative reducibility and o-machines rather than a-machines and most computing processes in the real world are potentially online or interactive. Therefore, we argue that Turing o-machines, relative computability, and online computing are the most important concepts in the subject, more so than Turing a-machines and standard computable functions since they are special cases of the former and are presented first only for pedagogical clarity to beginning students. At the end in §10–§13 we consider three displacements in computability theory, and the historical reasons they occurred. Several brief conclusions are drawn in §14. (shrink)

We investigate the connections between computability theory and Nonstandard Analysis. In particular, we investigate the two following topics and show that they are intimately related. A basic property of Cantor space$2^ $ is Heine–Borel compactness: for any open covering of $2^ $, there is a finite subcovering. A natural question is: How hard is it to compute such a finite subcovering? We make this precise by analysing the complexity of so-called fan functionals that given any $G:2^ \to $, (...) output a finite sequence $\langle f_0, \ldots,f_n \rangle $ in $2^ $ such that the neighbourhoods defined from $\overline {f_i } G\left$ for $i \le n$ form a covering of $2^ $. A basic property of Cantor space in Nonstandard Analysis is Abraham Robinson’s nonstandard compactness, i.e., that every binary sequence is “infinitely close” to a standard binary sequence. We analyse the strength of this nonstandard compactness property of Cantor space, compared to the other axioms of Nonstandard Analysis and usual mathematics.Our study of yields exotic objects in computability theory, while leads to surprising results in Reverse Mathematics. We stress that and are highly intertwined, i.e., our study is holistic in nature in that results in computability theory yield results in Nonstandard Analysis and vice versa. (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 (...) class='Hi'>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)

The paper is a contribution to intuitionistic reverse mathematics. We introduce a formal system called Basic Intuitionistic Mathematics BIM, and then search for statements that are, over BIM, equivalent to Brouwer’s Fan Theorem or to its positive denial, Kleene’s Alternative to the Fan Theorem. The Fan Theorem is true under the intended intuitionistic interpretation and Kleene’s Alternative is true in the model of BIM consisting of the Turing-computable functions. The task of finding equivalents of Kleene’s Alternative is, intuitionistically, a nontrivial (...) extension of the task of finding equivalents of the Fan Theorem, although there is a certain symmetry in the arguments that we shall try to make transparent. We introduce closed-and-separable subsets of Baire space $${\mathcal{N}}$$ and of the set $${\mathcal{R}}$$ of the real numbers. Such sets may be compact and also positively noncompact. The Fan Theorem is the statement that Cantor space $${\mathcal{C}}$$, or, equivalently, the unit interval [0, 1], is compact and Kleene’s Alternative is the statement that $${\mathcal{C}}$$, or, equivalently, [0, 1], is positively noncompact. The class of the compact closed-and-separable sets and also the class of the closed-and-separable sets that are positively noncompact are characterized in many different ways and a host of equivalents of both the Fan Theorem and Kleene’s Alternative is found. (shrink)

The aim of this paper is to give a description of the free algebras in some varieties of Glivenko MTL-algebras having the Boolean retraction property. This description is given (generalizing the results of [9]) in terms of weak Boolean products over Cantor spaces. We prove that in some cases the stalks can be obtained in a constructive way from free kernel DL-algebras, which are the maximal radical of directly indecomposable Glivenko MTL-algebras satisfying the equation in the title. We include (...) examples to show how we can apply the results to describe free algebras in some well known varieties of involutive MTL-algebras and of pseudocomplemented MTL-algebras. (shrink)

In this paper we study the logical strength of the determinacy of infinite binary games in terms of second order arithmetic. We define new determinacy schemata inspired by the Wadge classes of Polish spaces and show the following equivalences over the system RCA0*, which consists of the axioms of discrete ordered semi‐rings with exponentiation, Δ10 comprehension and Π00 induction, and which is known as a weaker system than the popularbase theory RCA0: 1. Bisep(Δ10, Σ10)‐Det* ↔ WKL0, 2. Bisep(Δ10, Σ20)‐Det* ↔ (...) ATR0 + Σ11 induction, 3. Bisep(Σ10, Σ20)‐Det* ↔ Sep(Σ10, Σ20)‐Det* ↔ Π11‐CA0, 4. Bisep(Δ20, Σ20)‐Det* ↔ Π11‐TR0, where Det* stands for the determinacy of infinite games in the Cantor space (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

§1. I will start with a quick definition of descriptive set theory: It is the study of the structure of definable sets and functions in separable completely metrizable spaces. Such spaces are usually called Polish spaces. Typical examples are ℝn, ℂn, Hilbert space and more generally all separable Banach spaces, the Cantor space 2ℕ, the Baire space ℕℕ, the infinite symmetric group S∞, the unitary group, the group of measure preserving transformations of the unit interval, etc.In (...) this theory sets are classified in hierarchies according to the complexity of their definitions and the structure of sets in each level of these hierarchies is systematically analyzed. In the beginning we have the Borel sets in Polish spaces, obtained by starting with the open sets and closing under the operations of complementation and countable unions, and the corresponding Borel hierarchy. After this come the projective sets, obtained by starting with the Borel sets and closing under the operations of complementation and projection, and the corresponding projective hierarchy.There are also transfinite extensions of the projective hierarchy and even much more complex definable sets studied in descriptive set theory, but I will restrict myself here to Borel and projective sets, in fact just those at the first level of the projective hierarchy, i.e., the Borel (), analytic () and coanalytic () sets. (shrink)

We analyze the pointwise convergence of a sequence of computable elements of L1 in terms of algorithmic randomness. We consider two ways of expressing the dominated convergence theorem and show that, over the base theory RCA0, each is equivalent to the assertion that every Gδ subset of Cantor space with positive measure has an element. This last statement is, in turn, equivalent to weak weak Königʼs lemma relativized to the Turing jump of any set. It is also equivalent (...) to the conjunction of the statement asserting the existence of a 2-random relative to any given set and the principle of Σ2 collection. (shrink)

We show that if κ is a weakly compact cardinal then the embeddability relation on trees of size κ is invariantly universal. This means that for every analytic quasi-order R on the generalized Cantor space View the MathML source there is an Lκ+κ-sentence φ such that the embeddability relation on its models of size κ, which are all trees, is Borel bi-reducible to R. In particular, this implies that the relation of embeddability on trees of size κ is (...) complete for analytic quasi-orders on View the MathML source. These facts generalize analogous results for κ=ω obtained in Louveau and Rosendal [17] and Friedman and Motto Ros [6], and it also partially extends a result from Baumgartner [3] concerning the structure of the embeddability relation on linear orders of size κ. (shrink)

The completeness of the modal logic S4 for all topological spaces as well as for the real line , the n-dimensional Euclidean space and the segment etc. was proved by McKinsey and Tarski in 1944. Several simplified proofs contain gaps. A new proof presented here combines the ideas published later by G. Mints and M. Aiello, J. van Benthem, G. Bezhanishvili with a further simplification. The proof strategy is to embed a finite rooted Kripke structure for S4 into a (...) subspace of the Cantor space which in turn encodes . This provides an open and continuous map from onto the topological space corresponding to . The completeness follows as S4 is complete with respect to the class of all finite rooted Kripke structures. (shrink)

If $\mathcal{F}$ is a filter on $\omega$, we say that $\mathcal{F}$ is Canjar if the corresponding Mathias forcing does not add a dominating real. We prove that any Borel Canjar filter is $F_{\sigma}$, solving a problem of Hrušák and Minami. We give several examples of Canjar and non-Canjar filters; in particular, we construct a $\mathsf{MAD}$ family such that the corresponding Mathias forcing adds a dominating real. This answers a question of Brendle. Then we prove that in all the “classical” models (...) of $\mathsf{ZFC}$ there are $\mathsf{MAD}$ families whose Mathias forcing does not add a dominating real. We also study ideals generated by branches, and we uncover a close relation between Canjar ideals and the selection principle $S_{\mathrm{fin}}$ on subsets of the Cantor space. (shrink)

This paper studies, by way of an example, the intuitionistic propositional connective * defined in the language of second order propositional logic by * ≡ ∃Q. In full topological models * is not generally definable but over Cantor-space and the reals it can be classically shown that *↔ ⅂⅂P; on the other hand, this is false constructively, i.e. a contradiction with Church's thesis is obtained. This is comparable with some well-known results on the completeness of intuitionistic first-order predicate (...) logic. Over [0, 1], the operator * is undefinable. We show how to recast this argument in terms of intuitive intuitionistic validity in some parameter. The undefinability argument essentially uses the connectedness of [0, 1]; most of the work of recasting consists in the choice of a suitable intuitionistically meaningful parameter, so as to imitate the effect of connectedness. Parameters of the required kind can be obtained as so-called projections of lawless sequences. (shrink)

We show that there is a computable procedure which, given an ∀∃-sentence ${\varphi}$ in the language of the partially ordered sets with a top element 1 and a bottom element 0, computes whether ${\varphi}$ is true in the Medvedev degrees of ${\Pi^0_1}$ classes in Cantor space, sometimes denoted by ${\mathcal{P}_s}$.