It is known that the box dimension of any Martin-Löf random closed set of ${\{0,1\}^\mathbb{N}}$ is ${\log_2(\frac{4}{3})}$ . Barmpalias et al. [J Logic Comput 17(6):1041–1062, 2007] gave one method of producing such random closed sets and then computed the box dimension, and posed several questions regarding other methods of construction. We outline a method using random recursive constructions for computing the Hausdorff dimension of almost every random closed set of ${\{0,1\}^\mathbb{N}}$ , and (...) propose a general method for random closed sets in other spaces. We further find both the appropriate dimensional Hausdorff measure and the exact Hausdorff dimension for such random closed sets. (shrink)

We show that if a real x2ω is strongly Hausdorff -random, where h is a dimension function corresponding to a convex order, then it is also random for a continuous probability measure μ such that the μ-measure of the basic open cylinders shrinks according to h. The proof uses a new method to construct measures, based on effective continuous transformations and a basis theorem for -classes applied to closed sets of probability measures. We use the main (...) result to derive a collapse of randomness notions for Hausdorff measures, and to provide a characterization of effective Hausdorff dimension similar to Frostman’s Theorem. (shrink)

We study inversions of the jump operator on ${\mathrm{\Pi }}_{1}^{0}$ classes, combined with certain basis theorems. These jump inversions have implications for the study of the jump operator on the random degrees—for various notions of randomness. For example, we characterize the jumps of the weakly 2-random sets which are not 2-random, and the jumps of the weakly 1-random relative to 0′ sets which are not 2-random. Both of the classes coincide with the degrees (...) above 0′ which are not 0′-dominated. A further application is the complete solution of [24, Problem 3.6.9]: one direction of van Lambalgen's theorem holds for weak 2-randomness, while the other fails. Finally we discuss various techniques for coding information into incomplete randoms. Using these techniques we give a negative answer to [24, Problem 8.2.14]: not all weakly 2-random sets are array computable. In fact, given any oracle X, there is a weakly 2-random which is not array computable relative to X. This contrasts with the fact that all 2-random sets are array computable. (shrink)

This work contributes to the program of studying effective versions of “almost-everywhere” theorems in analysis and ergodic theory via algorithmic randomness. Consider the setting of Cantor space {0,1}N with the uniform measure and the usual shift. We determine the level of randomness needed for a point so that multiple recurrence in the sense of Furstenberg into effectively closed sets P of positive measure holds for iterations starting at the point. This means that for each k∈N there is an (...) n such that n,2n,…,kn shifts of the point all end up in P. We consider multiple recurrence into closed sets that possess various degrees of effectiveness: clopen, Π10 with computable measure, and Π10. The notions of Kurtz, Schnorr, and Martin-Löf randomness, respectively, turn out to be sufficient. We obtain similar results for multiple recurrence with respect to the k commuting shift operators on {0,1}Nk. (shrink)

Simpson introduced the lattice of Π0 1 classes under Medvedev reducibility. Questions regarding completeness in are related to questions about measure and randomness. We present a solution to a question of Simpson about Medvedev degrees of Π0 1 classes of positive measure that was independently solved by Simpson and Slaman. We then proceed to discuss connections to constructive logic. In particular we show that the dual of does not allow an implication operator (i.e. that is not a Heyting algebra). We (...) also discuss properties of the class of PA-complete sets that are relevant in this context. (shrink)

An illustrated exploration of fandom that combines academic essays with artist pages and experimental texts. Fandom as Methodology examines fandom as a set of practices for approaching and writing about art. The collection includes experimental texts, autobiography, fiction, and new academic perspectives on fandom in and as art. Key to the idea of “fandom as methodology” is a focus on the potential for fandom in art to create oppositional spaces, communities, and practices, particularly from queer perspectives, but also through transnational, (...) feminist and artist-of-color fandoms. The book provides a range of examples of artists and writers working in this vein, as well as academic essays that explore the ways in which fandom can be theorized as a methodology for art practice and art history. Fandom as Methodology proposes that many artists and art writers already draw on affective strategies found in fandom. With the current focus in many areas of art history, art writing, and performance studies around affective engagement with artworks and imaginative potentials, fandom is a key methodology that has yet to be explored. Interwoven into the academic essays are lavishly designed artist pages in which artists offer an introduction to their use of fandom as methodology. Contributors Taylor J. Acosta, Catherine Grant, Dominic Johnson, Kate Random Love, Maud Lavin, Owen G. Parry, Alice Butler, SooJin Lee, Jenny Lin, Judy Batalion, Ika Willis. Artists featured in the artist pages Jeremy Deller, Ego Ahaiwe Sowinski, Anna Bunting-Branch, Maria Fusco, Cathy Lomax, Kamau Amu Patton, Holly Pester, Dawn Mellor, Michelle Williams Gamaker, The Women of Colour Index Reading Group, Liv Wynter, Zhiyuan Yang. (shrink)

Much of the commentary in the wake of the Charlie Gard litigation was aimed at apparent shortcomings of the law. These include concerns about the perceived inability of the law to consider resourcing issues, the vagueness of the best interests test and the delays and costs of having disputes about potentially life-sustaining medical treatment resolved by the courts. These concerns are perennial ones that arise in response to difficult cases. Despite their persistence, we argue that many of these criticisms are (...) unfounded. The first part of this paper sets out the basic legal framework that operates when parents seek potentially life-sustaining treatment that doctors believe is against a child’s best interests, and describes the criticisms of that framework. The second part of the paper suggests an alternative approach that would give decision-making power to parents, and remove doctors’ ability to unilaterally withhold or withdraw life-sustaining treatment that they regard is futile. This proposal is grounded in several values that we argue should guide these regulatory choices. We also contend that the best interests test is justifiable and since the courts show no sign of departing from it, the focus should be on how to better elucidate the underlying values driving decisions. We discuss the advantages of our proposed approach and how it would address some of the criticisms aimed at the law. Finally, we defend the current role that the judiciary plays, as an independent state-sanctioned process with a precedent-setting function. (shrink)

PARC is an "appended numeral" system of natural deduction that I learned as an undergraduate and have taught for many years. Despite its considerable pedagogical strengths, PARC appears to have never been published. The system features explicit "tracking" of premises and assumptions throughout a derivation, the collapsing of indirect proofs into conditional proofs, and a very simple set of quantificational rules without the long list of exceptions that bedevil students learning existential instantiation and universal generalization. The system can be used (...) with any Copi-style set of inference rules, so it is quite adaptable to many mainstream symbolic logic textbooks. Consequently, PARC may be especially attractive to logic teachers who find Jaskowski/Gentzen-style introduction/elimination rules to be far less "natural" than Copi-style rules. The PARC system is also keyboard-friendly in comparison to the widely adopted Jaskowski-style graphical subproof system of natural deduction, viz., Fitch diagrams and Copi "bent arrow" diagrams. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Anthony Close CENTERING THE DE-CENTERERS: FOUCAULT AND LAS MENINAS Over the last two decades, French avant-garde critical theory has shaken the pillars of the traditionalist temple with this thought: the interpreter of a literary text should not primarily be concerned with its author's intentional design, but rather with the surreptitious forces which shape it, warp it, and ultimately turn it into a problematic will-ofthe -wisp. The "decoding" of those (...) forces takes many forms (Freudian, anthropological, and so on), of which deconstruction is the most celebrated ; "de-centering"—the banishment of the intentional subject from the focus of interpretation — is common to all. In the first part of this article I wish to set out my philosophical objections to that exclusion, and in the second part, to examine an interesting failure to make it effective in practice: Foucault's analysis of Velazquez's painting Las Meninas, which constitutes the first chapter of Les mots et ks choses. One of my reasons for choosing Foucault's text as an example is that its premises are in some ways comparable to the philosophy of the archdeconstructor, Derrida. My skepticism about them and the analytic method based on them is not meant to suggest that I regard Foucault's historical studies as valueless. On the contrary, I recognize that he, together with some other decoders (particularly Barthes), has important things to say. My attack is a form of defense. I wish to uphold the value of traditional forms of interpretation by showing that though Foucault, like other decoders, casts radical doubts upon them in theory, he tends to negate those doubts by the implications of his practice. 21 22Philosophy and Literature M. H. Abrams has defined the traditionalist historian of culture as one who assumes that (a) authors of the past exploited the norms and possibilities of their language to say something determinate, (b) the properly equipped historian can approximate to their meaning, (c) his interpretation can aspire to a reasonable degree of objectivity. ' Let that serve as a working definition of my traditionalist standpoint. This has a philosophical rationale, basically Wittgensteinian and Austinian, which needs to be explained and set over against some of decoding's typical presuppositions. I choose Derrida's as an example. From my perspective, two objections can be leveled against his thought: its tendency to hypostatize the intentional subject as a metaphysical essence which must be reached through and beyond language, and to see language itself as an unrealistically slippery medium. Both objections were raised against him by Searle in their famous polemic with each other.2 Derrida's thought is inscribed within, even while it reacts against, a traditional conception of language as a name-like substitute for essence. According to this conception, influentially represented in this century by Husserl's phenomenology, bits of language somewhat resemble sterling currency bills which bear on their face the legend: "I promise to pay the bearer on demand the sum of? pound(s)" That is, they are potentially capable ofbeing cashed for lumps of being, which can thus be possessed by mind. Derrida counters that the currency is fraudulent: it continually defers presence and possession, rendering problematic its relation to being and self, and their very existence. In order to foster the illusion that he is master of his words, hence of what they designate, Western Man has historically given status of originality to voice rather than to writing, to a form of expression which is seemingly intimate to consciousness rather than external. The paradoxical thesis ofDe lagrammatologie is that writing came first. One of the routes by which Derrida arrives at this conclusion is Saussure's conception of the discontinuous, articulated nature of language and the negative and differential character of linguistic signifiers, not stably linked to universal referents.3 On this basis he builds his idea of différance (a pun on difference and deferral ), the elusive negative margin which makes language possible, and at the same time, makes impossible consciousness's full presence to itself and to being.4 Hence Derrida's notion of language's anarchic and willful power to detach itself from die intentions or usage of its users and to float... (shrink)

We investigate notions of randomness in the space ${{\mathcal C}(2^{\mathbb N})}$ of continuous functions on ${2^{\mathbb N}}$ . A probability measure is given and a version of the Martin-Löf test for randomness is defined. Random ${\Delta^0_2}$ continuous functions exist, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set and hence uncountable. For any ${y (...) \in 2^{\mathbb N}}$ , there exists a random continuous function F with y in the image of F. Thus the image of a random continuous function need not be a random closed set. The set of zeroes of a random continuous function is always a random closed set. (shrink)

Let ≤r and ≤sbe two binary relations on 2ℕ which are meant as reducibilities. Let both relations be closed under finite variation and consider the uniform distribution on 2ℕ, which is obtained by choosing elements of 2ℕ by independent tosses of a fair coin.Then we might ask for the probability that the lower ≤r-cone of a randomly chosen set X, that is, the class of all sets A with A ≤rX, differs from the lower ≤s-cone of X. By (...) c osure under finite variation, the Kolmogorov 0-1 aw yields immediately that this probability is either 0 or 1; in case it is 1, the relations are said to be separable by random oracles.Again by closure under finite variation, for every given set A, the probability that a randomly chosen set X is in the upper ≤r-cone of A is either 0 or 1; let Almostr be the class of sets for which the upper ≤r-cone has measure 1. In the following, results about separations by random oracles and about Almost classes are obtained in the context of generalized reducibilities, that is, for binary relations on 2ℕ which can be defined by a countable set of total continuous functionals on 2ℕ in the same way as the usual resource-bounded reducibilities are defined by an enumeration of appropriate oracle Turing machines. The concept of generalized reducibility comprises a natura resource-bounded reducibilities, but is more general; in particular, it does not involve any kind of specific machine model or even effectivity. The results on generalized reducibilities yield corollaries about specific resource-bounded reducibilities, including several results which have been shown previously in the setting of time or space bounded Turing machine computations. (shrink)

It is time for a new paper about open questions in the currently very active area of randomness and computability. Ambos-Spies and Kučera presented such a paper in 1999 [1]. All the question in it have been solved, except for one: is KL-randomness different from Martin-Löf randomness? This question is discussed in Section 6.Not all the questions are necessarily hard—some simply have not been tried seriously. When we think a question is a major one, and therefore likely to be hard, (...) we indicate this by the symbol ▶, the criterion being that it is of considerable interest and has been tried by a number of researchers. Some questions are close contenders here; these are marked by ▷. With few exceptions, the questions are precise. They mostly have a yes/no answer. However, there are often more general questions of an intuitive or even philosophical nature behind. We give an outline, indicating the more general questions.All sets will be sets of natural numbers, unless otherwise stated. These sets are identified with infinite strings over {0, 1}. Other terms used in the literature are sequence and real. (shrink)

Background Voluntary assisted dying became lawful in Victoria, the first Australian state to permit this practice, in 2019 via the Voluntary Assisted Dying Act 2017 (Vic). While conscientious objection by individual health professionals is protected by the Victorian legislation, objections by institutions are governed by policy. No research has been conducted in Victoria, and very little research conducted internationally, on how institutional objection is experienced by patients seeking assisted dying. Methods 28 semi-structured interviews were conducted with 32 family caregivers and (...) one patient about the experience of 28 patients who sought assisted dying. Participants were interviewed during August-November 2021. Data from the 17 interviews (all with family caregivers) which reported institutional objection were analysed thematically. Results Participants reported institutional objection affecting eligibility assessments, medication access, and taking the medication or having it administered. Institutional objection occurred across health settings and was sometimes communicated obliquely. These objections resulted in delays, transfers, and choices between progressing an assisted dying application and receiving palliative or other care. Participants also reported objections causing adverse emotional experiences and distrust of objecting institutions. Six mediating influences on institutional objections were identified: staff views within objecting institutions; support of external medical practitioners and pharmacists providing assisted dying services; nature of a patient’s illness; progression or state of a patient’s illness; patient’s geographical location; and the capability and assertiveness of a patient and/or caregiver. Conclusions Institutional objection to assisted dying is much-debated yet empirically understudied. This research found that in Victoria, objections were regularly reported by participants and adversely affected access to assisted dying and the wider end-of-life experience for patients and caregivers. This barrier arises in an assisted dying system that is already procedurally challenging, particularly given the limited window patients have to apply. Better regulation may be needed as Victoria’s existing policy approach appears to preference institutional positions over patient’s choice given existing power dynamics. (shrink)

The interplay between computability and randomness has been an active area of research in recent years, reflected by ample funding in the USA, numerous workshops, and publications on the subject. The complexity and the randomness aspect of a set of natural numbers are closely related. Traditionally, computability theory is concerned with the complexity aspect. However, computability theoretic tools can also be used to introduce mathematical counterparts for the intuitive notion of randomness of a set. Recent research shows that, conversely, concepts (...) and methods originating from randomness enrich computability theory.The book covers topics such as lowness and highness properties, Kolmogorov complexity, betting strategies and higher computability. Both the basics and recent research results are desribed, providing a very readable introduction to the exciting interface of computability and randomness for graduates and researchers in computability theory, theoretical computer science, and measure theory. (shrink)

We consider the question of randomness of the probability ΩU[X] that an optimal Turing machine U halts and outputs a string in a fixed set X. The main results are as follows: ΩU[X] is random whenever X is $\Sigma _{n}^{0}$-complete or $\Pi _{n}^{0}$-complete for some n ≥ 2. However, for n ≥ 2, ΩU[X] is not n-random when X is $\Sigma _{n}^{0}$ or $\Pi _{n}^{0}$ Nevertheless, there exists $\Delta _{n+1}^{0}$ sets such that ΩU[X] is n-random. There (...) are $\Delta _{2}^{0}$ sets X such that ΩU[X] is rational. Also, for every n ≥ 1, there exists a set X which is $\Delta _{n+1}^{0}$ and $\Sigma _{n}^{0}$-hard such that ΩU[X] is not random. We also look at the range of ΩU as an operator. We prove that the set {ΩU[X]: X ⊆ 2<ω} is a finite union of closed intervals. It follows that for any optimal machine U and any sufficiently small real r, there is a set X ⊆ 2<ω recursive in ∅′ ⊕ r, such that ΩU[X] = r. The same questions are also considered in the context of infinite computations, and lead to similar results. (shrink)

A total of 1918 loci, detected by the hybridization of 938 expressed sequence tag unigenes from 26 Triticeae cDNA libraries, were mapped to wheat homoeologous group 4 chromosomes using a set of deletion, ditelosomic, and nulli-tetrasomic lines. The 1918 EST loci were not distributed uniformly among the three group 4 chromosomes; 41, 28, and 31% mapped to chromosomes 4A, 4B, and 4D, respectively. This pattern is in contrast to the cumulative results of EST mapping in all homoeologous groups, as reported (...) elsewhere, that found the highest proportion of loci mapped to the B genome. Sixty-five percent of these 1918 loci mapped to the long arms of homoeologous group 4 chromosomes, while 35% mapped to the short arms. The distal regions of chromosome arms showed higher numbers of loci than the proximal regions, with the exception of 4DL. This study confirmed the complex structure of chromosome 4A that contains two reciprocal translocations and two inversions, previously identified. An additional inversion in the centromeric region of 4A was revealed. A consensus map for homoeologous group 4 was developed from 119 ESTs unique to group 4. Forty-nine percent of these ESTs were found to be homoologous to sequences on rice chromosome 3, 12% had matches with sequences on other rice chromosomes, and 39% had no matches with rice sequences at all. Limited homology was found between wheat ESTs on homoeologous group 4 and the Arabidopsis genome. Forty-two percent of the homoeologous group 4 ESTs could be classified into functional categories on the basis of blastX searches against all protein databases. (shrink)

A real x is -Kurtz random if it is in no closed null set . We show that there is a cone of -Kurtz random hyperdegrees. We characterize lowness for -Kurtz randomness as being -dominated and -semi-traceable.

A seminal theorem due to Weyl [14] states that if $(a_n)$ is any sequence of distinct integers, then, for almost every $x \in \mathbb{R}$, the sequence $(a_n x)$ is uniformly distributed modulo one. In particular, for almost every $x$ in the unit interval, the sequence $(a_n x)$ is uniformly distributed modulo one for every computable sequence $(a_n)$ of distinct integers. Call such an $x$ UD random. Here it is shown that every Schnorr random real is UD random, (...) but there are Kurtz random reals that are not UD random. On the other hand, Weyl's theorem still holds relative to a particular effectively closed null set, so there are UD random reals that are not Kurtz random. (shrink)

This report describes the rationale, approaches, organization, and resource development leading to a large-scale deletion bin map of the hexaploid wheat genome. Accompanying reports in this issue detail results from chromosome bin-mapping of expressed sequence tags representing genes onto the seven homoeologous chromosome groups and a global analysis of the entire mapped wheat EST data set. Among the resources developed were the first extensive public wheat EST collection. Described are protocols for sequencing, sequence processing, EST nomenclature, and the assembly of (...) ESTs into contigs. These contigs plus singletons were used for selection of distinct sequence motif unigenes. Selected ESTs were rearrayed, validated by 5′ and 3′ sequencing, and amplified for probing a series of wheat aneuploid and deletion stocks. Images and data for all Southern hybridizations were deposited in databases and were used by the coordinators for each of the seven homoeologous chromosome groups to validate the mapping results. Results from this project have established the foundation for future developments in wheat genomics. (shrink)

We reformulate slightly Russell's notion of typicality, so as to eliminate its circularity and make it applicable to elements of any first‐order structure. We argue that the notion parallels Martin‐Löf (ML) randomness, in the sense that it uses definable sets in place of computable ones and sets of “small” cardinality (i.e., strictly smaller than that of the structure domain) in place of measure zero sets. It is shown that if the domain M satisfies, then there exist typical (...) elements and only non‐typical ones. In particular this is true for the standard model of second‐order arithmetic. By allowing parameters in the defining formulas, we are led to relative typicality, which satisfies most of van Lambalgen's axioms for relative randomness. However van Lambalgen's theorem is false for relative typicality. The class of typical reals is incomparable (with respect to ⊆) with the classes of ML‐random, Schnorr random and computably random reals. Also the class of typical reals is closed under Turing degrees and under the jump operation (both ways). (shrink)

Kendall's (1956) approach to the general epidemic is generalized by dropping the assumptions of constant infectivity and random recovery or death of ill individuals. A great deal of attention is paid to the biological background and the heuristics of the model formulation. Some new results are: (l) the derivation of Kermack's and McKendrick's integral equation from what seems to be the most general set of assumptions in section 2.2, (2) the use of Kermack's and McKendrick's final value equation to (...) arrive at a finite time version of the threshold theorem for the general case, comparable to that for the case of only one Markovian state of illness in section 2.5, (3) the analysis of the behaviour of the solutions of the integral equation when the starting infection approaches zero in section 2.7, (4) the derivation of the probability structure of a general branching process, after conditioning on extinction in section 3.6, (5) the statement of the generalized versions of Kendall's ideas in the form of precise limit conjectures in section 4, (6) the derivation of a closed expression for the limit epidemic resulting from (3) in appendix 4. (shrink)

In this paper, we introduce and study the concept of" neutrosophic closed set" and" neutrosophic continuous function". Possible application to GIS topology rules are touched upon.

Let a trace be a computably enumerable set of natural numbers such that V[m]={n:〈n,m〉∈V}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V^{[m]} = \{n : \langle n, m\rangle \in V \}}$$\end{document} is finite for all m, where 〈.,.〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle^{.},^{.}\rangle}$$\end{document} denotes an appropriate pairing function. After looking at some basic properties of traces like that there is no uniform enumeration of all traces, we prove varied results on traceability and variants thereof, where (...) a function f:N→N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f : \mathbb{N} \rightarrow \mathbb{N}}$$\end{document} is traceable via a trace V if for all m,〈f(m),m〉∈V.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${m, \langle f(m), m\rangle \in V.}$$\end{document} Then we turn to lattices Ltr(V)=({W:V⊆WandWatrace},⊆),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{\textbf{L}}_{tr}(V) = (\{W : V \subseteq W \, {\rm and} \, W \, {\rm a} \, {\rm trace}\}, \, \subseteq),$$\end{document}V a trace. Here, we study the close relationship to E=({A:A⊆Nc.e.},⊆)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{E} = (\{A : A \subseteq \mathbb{N} \quad c.e.\}, \subseteq)}$$\end{document}, automorphisms, isomorphisms, and isomorphic embeddings. (shrink)

An effectively closed set, or ${\Pi^{0}_{1}}$ class, may viewed as the set of infinite paths through a computable tree. A numbering, or enumeration, is a map from ω onto a countable collection of objects. One numbering is reducible to another if equality holds after the second is composed with a computable function. Many commonly used numberings of ${\Pi^{0}_{1}}$ classes are shown to be mutually reducible via a computable permutation. Computable injective numberings are given for the family of ${\Pi^{0}_{1}}$ classes (...) and for the subclasses of decidable and of homogeneous ${\Pi^{0}_{1}}$ classes. However no computable numberings exist for small or thin classes. No computable numbering of trees exists that includes all computable trees without dead ends. (shrink)

We study the values of the higher dimensional cardinal characteristics for sets of functions $f:\omega ^\omega \to \omega ^\omega $ introduced by the second author in [8]. We prove that while the bounding numbers for these cardinals can be strictly less than the continuum, the dominating numbers cannot. We compute the bounding numbers for the higher dimensional relations in many well known models of $\neg \mathsf {CH}$ such as the Cohen, random and Sacks models and, as a byproduct (...) show that, with one exception, for the bounding numbers there are no $\mathsf {ZFC}$ relations between them beyond those in the higher dimensional Cichoń diagram. In the case of the dominating numbers we show that in fact they collapse in the sense that modding out by the ideal does not change their values. Moreover, they are closely related to the dominating numbers $\mathfrak {d}^\lambda _\kappa $. (shrink)

We study some properties of the quotient forcing notions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Q_{tr(I)} = \wp(2^{< \omega})/tr(i)}$$\end{document} and PI = B(2ω)/I in two special cases: when I is the σ-ideal of meager sets or the σ-ideal of null sets on 2ω. We show that the remainder forcing RI = Qtr(I)/PI is σ-closed in these cases. We also study the cardinal invariant of the continuum \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} (...) \begin{document}$${\mathfrak{h}_{\mathbb{Q}}}$$\end{document}, the distributivity number of the quotient \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Dense(\mathbb{Q})/nwd}$$\end{document}, in order to show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\wp(\mathbb{Q})/nwd}$$\end{document} collapses \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{c}}$$\end{document} to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{h}_{\mathbb{Q}}}$$\end{document}, thus answering a question addressed in Balcar et al. (Fundamenta Mathematicae 183:59–80, 2004). (shrink)

The neutrosophic sets were known since 1999, and because of their wide applications and their great flexibility to solve the problems, we used these the concepts to define a new types of neutrosophic crisp closed sets and limit points in neutrosophic crisp topological space, namly [neutrosophic crisp Gem sets and neutrosophic crisp Turig points] respactvely, we stady their properties in details and join it with topological concepts. Finally we used [neutrosophic crisp Gem sets and neutrosophic (...) crisp Turig points] to introduce of topological concepts as: neutrosophic crisp closed (open) sets, neutrosophic crisp closure, neutrosophic crisp interior, neutrosophic crisp extrior and neutrosophic crisp boundary which are fundamental for further reserch on neutrosophic crisp topology and will setrengthen the foundations of theory of neutrosophic topological spaces. (shrink)

If there is a homeomorphic embedding of one set into another, the sets are said to be topologically comparable. Friedman and Hirst have shown that the topological comparability of countable closed subsets of the reals is equivalent to the subsystem of second order arithmetic denoted byATR 0. Here, this result is extended to countable closed locally compact subsets of arbitrary complete separable metric spaces. The extension uses an analogue of the one point compactification of ℝ.

Let T be a one-based theory. We define a notion of width, in the case of T having the finiteness property, for the lattice of finitely generated algebraically closed sets and prove Theorem. Let T be one-based with the finiteness property. If T is of bounded width, then every type in T is nonorthogonal to a weight one type. If T is countable, the converse is true.

This paper contains a corrected proof that the statement “every non-empty closed subset of a compact complete separable metric space is separably closed” implies the arithmetical comprehension axiom of reverse mathematics.

Let κ and λ be infinite cardinals, F a filter on κ, and G a set of functions from κ to κ. The filter F is generated by G if F consists of those subsets of κ which contain the range of some element of G. The set G is $ -closed if it is closed in the $ -topology on κ κ. (In general, the $ -topology on IA has basic open sets all Π i∈ I (...) U i such that, for all $i \in I, U_i \subseteq A$ and $|\{i \in I: U_i \neq A\}| .) The primary question considered in this paper asks "Is there a uniform ultrafilter on κ which is generated by a closed set of functions?" (Closed means $ -closed.) We also establish the independence of two related questions. One is due to Carlson: "Does there exist a regular cardinal κ and a subtree T of $^{ such that the set of branches of T generates a uniform ultrafilter on κ?"; and the other is due to Pouzet: "For all regular cardinals κ, is it true that no uniform ultrafilter on κ is analytic?" We show that if κ is a singular, strong limit cardinal, then there is a uniform ultrafilter on κ which is generated by a closed set of increasing functions. Also, from the consistency of an almost huge cardinal, we get the consistency of CH + "There is a uniform ultrafilter on ℵ 1 which is generated by a closed set of increasing functions". In contrast with the above results, we show that if κ is any cardinal, λ is a regular cardinal less than or equal to κ and P is the forcing notion for adding at least $(\kappa^{ generic subsets of λ, then in V P , no uniform ultrafilter on κ is generated by a closed set of functions. (shrink)

In this paper, we establish a relationship between the join property and Turing degrees of members of effectively closed sets in Cantor space, i.e., classes. We first give a proof of the observation that there exists a non‐empty special class in which no join of two members computes the halting set. We then prove the existence of a non‐empty special class such that no member satisfies the join property, where a degree satisfies the join property if for all (...) non‐zero there exists such that. (shrink)

We show that the set of points of an overt closed subspace of a metric completion of a Bishop-locally compact metric space is located. Consequently, if the subspace is, moreover, compact, then its collection of points is Bishop-compact. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

The empty set of course contains no computable point. On the other hand, surprising results due to Zaslavskiĭ, Tseĭtin, Kreisel, and Lacombe have asserted the existence of non-empty co-r. e. closed sets devoid of computable points: sets which are even “large” in the sense of positive Lebesgue measure.This leads us to investigate for various classes of computable real subsets whether they always contain a computable point.

With every σ-ideal I on a Polish space we associate the σ-ideal I* generated by the closed sets in I. We study the forcing notions of Borel sets modulo the respective σ-ideals I and I* and find connections between their forcing properties. To this end, we associate to a σ-ideal on a Polish space an ideal on a countable set and show how forcing properties of the forcing depend on combinatorial properties of the ideal. We also study (...) the 1—1 or constant property of σ-ideals, i.e., the property that every Borel function defined on a Borel positive set can be restricted to a positive Borel set on which it either 1—1 or constant. We prove the following dichotomy: if I is a σ-ideal generated by closed sets, then either the forcing P I adds a Cohen real, or else I has the 1—1 or constant property. (shrink)

We prove that for every family I of closed subsets of a Polish space each Σ 1 1 set can be covered by countably many members of I or else contains a nonempty Π 0 2 set which cannot be covered by countably many members of I. We prove an analogous result for κ-Souslin sets and show that if A ♯ exists for any $A \subset \omega^\omega$ , then the above result is true for Σ 1 2 (...) class='Hi'>sets. A theorem of Martin is included stating that this result is also true for weakly homogeneously Souslin sets. As an application of our results we derive from them a general form of Hurewicz's theorem due to Kechris, Louveau, and Woodin and a theorem of Feng on the open covering axiom. Also some well-known theorems on finding "big" closed sets inside of "big" Σ 1 1 and Σ 1 2 sets are consequences of our results. (shrink)

It is well known that in Bishop-style constructive mathematics, the closure of the union of two subsets of ℝ is ‘not’ the union of their closures. The dual situation, involving the complement of the closure of the union, is investigated constructively, using completeness of the ambient space in order to avoid any application of Markov's Principle.

We discuss the problem of finding forcing posets which introduce closed unbounded subsets to a given stationary set.

We show that there is an antichain basis for neither the class of non-potentially closed Borel subsets of the plane under Borel rectangular reducibility nor the class of analytic graphs of uncountable Borel chromatic number under Borel reducibility.

EveryK-trivial set is computable from an incomplete Martin-Löf random set, i.e., a Martin-Löf random set that does not compute the halting problem.