This paper presents new results from a detailed study of the structure of autocatalytic sets. We show how autocatalytic sets can be decomposed into smaller autocatalytic subsets, and how these subsets can be identified and classified. We then argue how this has important consequences for the evolvability, enablement, and emergence of autocatalytic sets. We end with some speculation on how all this might lead to a generalized theory of autocatalytic sets, which (...) could possibly be applied to entire ecologies or even economies. (shrink)

This year we celebrated Stuart Kauffman’s 80th birthday. Kauffman has contributed many original ideas to science. One of them is that of autocatalytic sets in the context of the origin of life. An autocatalytic set is a self-sustaining chemical reaction network in which all the molecules mutually catalyze each other’s formation from a basic food source. This notion is often seen as a “counterargument” against the dominant genetics-first view of the origin of life, focusing more on metabolism (...) instead. The original notion was introduced back in 1971, but it has taken several decades for this idea to really catch on. Thanks to theoretical as well as experimental progress in more recent research on autocatalytic sets, especially over the past 15 years, the idea now seems to be gaining significant interest and support. In this tribute to Kauffman’s work and ideas, a brief history of research on autocatalytic sets is presented. (shrink)

Autocatalytic networks have been used to model the emergence of self‐organizing structure capable of sustaining life and undergoing biological evolution. Here, we model the emergence of cognitive structure capable of undergoing cultural evolution. Mental representations (MRs) of knowledge and experiences play the role of catalytic molecules, and interactions among them (e.g., the forging of new associations) play the role of reactions and result in representational redescription. The approach tags MRs with their source, that is, whether they were acquired through (...) social learning, individual learning (of pre‐existing information), or creative thought (resulting in the generation of new information). This makes it possible to model how cognitive structure emerges and to trace lineages of cumulative culture step by step. We develop a formal representation of the cultural transition from Oldowan to Acheulean tool technology using Reflexively Autocatalytic and Food set generated (RAF) networks. Unlike more primitive Oldowan stone tools, the Acheulean hand axe required not only the capacity to envision and bring into being something that did not yet exist, but hierarchically structured thought and action, and the generation of new MRs: the concepts EDGING, THINNING, SHAPING, and a meta‐concept, HAND AXE. We show how this constituted a key transition toward the emergence of semantic networks that were self‐organizing, self‐sustaining, and autocatalytic, and we discuss how such networks replicated through social interaction. The model provides a promising approach to unraveling one of the greatest anthropological mysteries: that of why development of the Acheulean hand axe was followed by over a million years of cultural stasis. (shrink)

The emergence of an autocatalytic network from an available set of elements is a fundamental step in early evolutionary processes, such as the origin of metabolism. Given the set of elements, the reactions between them (chemical or otherwise), and with various elements catalysing certain reactions, a Reflexively Autocatalytic F-generated (RAF) set is a subset R′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$'$$\end{document} of reactions that is self-generating from a given food set, and with each reaction in (...) R′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$'$$\end{document} being catalysed from within R′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$'$$\end{document}. RAF theory has been applied to various phenomena in theoretical biology, and a key feature of the approach is that it is possible to efficiently identify and classify RAFs within large systems. This is possible because RAFs can be described as the (nonempty) subsets of the reactions that are the fixed points of an (efficiently computable) interior map that operates on subsets of reactions. Although the main generic results concerning RAFs can be derived using just this property, we show that for systems with at least 12 reactions there are generic results concerning RAFs that cannot be proven using the interior operator property alone. (shrink)

An idea is not a replicator because it does not consist of coded self-assembly instructions. It may retain structure as it passes from one individual to another, but does not replicate it. The cultural replicator is not an idea but an associatively-structured network of them that together form an internal model of the world, or worldview. A worldview is a primitive, uncoded replicator, like the autocatalytic sets of polymers widely believed to be the earliest form of life. Primitive (...) replicators generate self-similar structure, but because the process happens in a piecemeal manner, through bottom-up interactions rather than a top-down code, they replicate with low fidelity, and acquired characteristics are inherited. Just as polymers catalyze reactions that generate other polymers, the retrieval of an item from memory can in turn trigger other items, thus cross-linking memories, ideas, and concepts into an integrated conceptual structure. Worldviews evolve idea by idea, largely through social exchange. An idea participates in the evolution of culture by revealing certain aspects of the worldview that generated it, thereby affecting the worldviews of those exposed to it. If an idea influences seemingly unrelated fields this does not mean that separate cultural lineages are contaminating one another, because it is worldviews, not ideas, that are the basic unit of cultural evolution. (shrink)

The improbability of a spontaneously generated self-assembling molecule has suggested that life began with a set of simpler, collectively replicating elements, such as an enclosed autocatalytic set of polymers (or autocell). Since replication occurs without a self-assembly code, acquired characteristics are inherited. Moreover, there is no strict distinction between alive and dead; one can only infer that an autocell was alive if it replicates. These features of early life render natural selection inapplicable to the description of its change-of-state because (...) they defy its underlying assumptions. Moreover, natural selection describes only randomly generated novelty; it cannot describe the emergence of form at the interface between organism and environment. Self-organization is also inadequate because it is restricted to interactions amongst parts; it too cannot account for context-driven change. A modified version of selection theory or self-organization would not work because the description of change-of-state through interaction with an incompletely specified context has a completely different mathematical structure, i.e. entails a non-Kolmogorovian probability model. It is proposed that the evolution of early life is appropriately described as lineage transformation through context-driven actualization of potential, with self-organized change-of-state being a special case of no contextual influence, and competitive exclusion of less fit individuals through a selection-like process possibly (but not necessarily) playing a secondary role. It is argued that natural selection played an important role in evolution only after genetically mediated replication was established. (shrink)

The goal of synthetic biology is to create artificial organisms. To achieve this it is essential to understand what life is. Metabolism-replacement systems, or (M, R)-systems, constitute a theory of life developed by Robert Rosen, characterized in the statement that organisms are closed to efficient causation, which means that they must themselves produce all the catalysts they need. This theory overlaps in part with other current theories, including autopoiesis, the chemoton, and autocatalytic sets, all of them invoking some (...) idea of closure. A simple model of an (M, R)-system has been implemented in the computer, and behaves in ways that may shed light on the requirements for a prebiotic self-organizing system. In addition to a trivial steady state in which nothing happens, it can establish a non-trivial steady state in which all intermediates have finite concentrations, with their rates of degradation balanced by their rates of synthesis. The system can be regenerated from the set of food components plus a single intermediate, and maintain itself in that state indefinitely, despite continuous degradation. At the very low compartment volumes that may have existed in prebiotic conditions, for example in cavities in minerals, or in micelles formed by simple amphiphiles, statistical fluctuations in the numbers of molecules need to be taken into account. With the stochastic approach there is no non-trivial steady state in strict mathematical terms, because the system will always collapse to the trivial state after sufficient time. However, the average time before collapse is so long for volumes greater than $10^{-19}\;{\textsc{l}}$ (much smaller than the volume of the order of $10^{-15}\;{\textsc{l}}$ for a typical bacterial cell) that for practical purposes the self-maintaining state of non-null concentrations becomes significant, recalling the situation of bistability that is observed in deterministic analysis. In turn, there exists a minimum size below which the self-organizing system cannot maintain itself on chemically relevant time scales. The value of the critical volume depends on the particular concentrations and rate constants assumed, but the principle could apply generally. (shrink)

The improbability of a spontaneously generated self-assembling molecule has suggested that life began with a set of simpler, collectively replicating elements, such as an enclosed autocatalytic set of polymers (or protocell). Since replication occurs without a self-assembly code, acquired characteristics are inherited. Moreover, there is no strict distinction between alive and dead; one can only infer that a protocell was alive if it replicates. These features of early life render natural selection inapplicable to the description of its change-of-state because (...) they defy its underlying assumptions. Moreover, natural selection describes only randomly generated novelty; it cannot describe the emergence of form at the interface between organism and environment. Self-organization is also inadequate because it is restricted to interactions amongst parts; it too cannot account for context-driven change. A modified version of selection theory or self-organization would not work because the description of change-of-state through interaction with an incompletely specified context has a completely different mathematical structure, i.e. entails a non-Kolmogorovian probability model. It is proposed that the evolution of early life is appropriately described as lineage transformation through context-driven actualization of potential (CAP), with self-organized change-of-state being a special case of no contextual influence, and competitive exclusion of less fit individuals through a selection-like process possibly (but not necessarily) playing a secondary role. It is argued that natural selection played an important role in evolution only after genetically mediated replication was established. (shrink)

This essay presents a scenario of the origin of life that is based on analysis of biological architecture and mechanical design at the microstructural level. My thesis is that the same architectural and energetic constraints that shape cells today also guided the evolution of the first cells and that the molecular scaffolds that support solid-phase biochemistry in modern cells represent living microfossils of past life forms. This concept emerged from the discovery that cells mechanically stabilize themselves using tensegrity architecture and (...) that these same building rules guide hierarchical self-assembly at all size scales (Sci Amer 278:48–57;1998). When combined with other fundamental design principles (e.g., energy minimization, topological constraints, structural hierarchies, autocatalytic sets, solid-state biochemistry), tensegrity provides a physical basis to explain how atomic and molecular elements progressively self-assembled to create hierarchical structures with increasingly complex functions, including living cells that can self-reproduce. BioEssays 22:1160–1170, 2000. © 2000 John Wiley & Sons, Inc. (shrink)

An idea is not a replicator because it does not consist of coded self-assembly instructions. It may retain structure as it passes from one individual to another, but does not replicate it. The cultural replicator is not an idea but an associatively-structured network of them that together form an internal model of the world, or worldview. A worldview is a primitive, uncoded replicator, like the autocatalytic sets of polymers widely believed to be the earliest form of life. Primitive (...) replicators generate self-similar structure, but because the process happens in a piecemeal manner, through bottom-up interactions rather than a top-down code, they replicate with low fidelity, and acquired characteristics are inherited. Just as polymers catalyze reactions that generate other polymers, the retrieval of an item from memory can in turn trigger other items, thus cross-linking memories, ideas, and concepts into an integrated conceptual structure. Worldviews evolve idea by idea, largely through social exchange. An idea participates in the evolution of culture by revealing certain aspects of the worldview that generated it, thereby affecting the worldviews of those exposed to it. If an idea influences seemingly unrelated fields this does not mean that separate cultural lineages are contaminating one another, because it is worldviews, not ideas, that are the basic unit of cultural evolution. (shrink)

This paper uses autocatalytic networks to model discontinuous cultural transitions involving cross-domain transfer, using as an illustrative example, artworks inspired by the oldest-known uncontested example of figurative art: the carving of the Hohlenstein-Stadel Löwenmensch, or lion-human. Autocatalytic networks provide a general modeling setting in which nodes are not just passive transmitters of activation; they actively galvanize, or “catalyze” the synthesis of novel nodes from existing ones This makes them uniquely suited to model how new structure grows out of (...) earlier structure, i.e., cumulative, generative network growth. They have been used to model the origin and early evolution of biological life, and the emergence of cognitive structures capable of undergoing cultural evolution. We conducted a study in which six individual creators and one group generated music, prose, poetry, and visual art inspired by the Hohlenstein-Stadel Löwenmensch, and answered questions about the process. The data revealed four through-lines by which they expressed the Löwenmensch in an alternative art form: lion-human hybrid, subtracting from the whole to reveal the form within, deterioration, and waiting to be found with a story to tell. Autocatalytic networks were used to model how these four spontaneously derived through-lines form a cultural lineage from Löwenmensch to artist to audience. We used the resulting data from three creators to model the cross-domain transfer from inspirational source to creative product. These four spontaneously-generated threads of cultural continuity formed the backbone of this Löwenmensch-inspired cultural lineage, enabling culture to evolve even in the face of discontinuity at the level conventional categories or domains. We know of no other theory of cultural evolution that accommodates cross-domain transfer or other forms of discontinuity. The approach paves the way for a broad scientific framework for the origins of evolutionary processes. (shrink)

Many in science are disposed not to take biosemiotics seriously, dismissing it as too anthropomorphic. Furthermore, biosemiotic apologetics are cast in top-down fashion, thereby adding to widespread skepticism. An effective response might be to approach biosemiotics from the bottom up, but the foundational assumptions that support Enlightenment science make that avenue impossible. Considerations from ecosystem studies reveal, however, that those conventional assumptions, although once possessing great utilitarian value, have come to impede deeper understanding of living systems because they implicitly depict (...) the evolution of the universe backward. Ecological dynamics suggests instead a smaller set of countervailing postulates that allows evolution to play forward and sets the stage for tripartite causalities, signs, and interpreters—the key elements of biosemiosis—to emerge naturally out of the interaction of chance with configurations of autocatalytic processes. Biosemiosis thereby appears as a fully legitimate outgrowth of the new metaphysic and shows promise for becoming the supervenient focus of a deeper perspective on the phenomenon of life. (shrink)

It is commonly presumed that abiotic membranes were colonized by proteins later on. Yet, hydrophobic peptides could have formed primordial protein‐dominated membranes on their own. In a metabolism‐first context, “autocatalytically closed” sets of statistical peptides could organize a self‐maintaining protometabolism, assisted by an unfolding set of ribotide‐related cofactors. Pairwise complementary ribotide cofactors may have formed docking guides for stochastic peptide formation, before replicating RNA emerged from this subset. Tidally recurring wet‐drying cycles and an early onset of photosynthetic activities are (...) considered most likely to meet the thermodynamic requirements. Conceivably, the earliest peptide‐dominated vesicles were engaged in light harvesting, together with isoprenoid‐tethered pigments, rather than providing an external boundary. Early on, the bulk of prebiotic organic matter can have formed a contiguous layer covering the mineral sediment, held in place by colloidal coherence of a hydrogel matrix. This unconventional scenario assumes a late onset of cellular individualization – perhaps from within, resembling endosporogenesis. (shrink)

This paper considers questions of continuity and change in education from the perspective of complexity theory, introducing the field to educationists who might not be familiar with it. Given a significant degree of complexity in a particular environment , new properties and behaviours, which are not necessarily contained in the essence of the constituent elements or able to be predicted from a knowledge of initial conditions, will emerge. These concepts of emergent phenomena from a critical mass, associated with notions of (...) lock‐in, path dependence, and inertial momentum, suggest that it is in the dynamic interactions and adaptive orientation of a system that new phenomena, new properties and behaviours, emerge. The focus thus shifts from a concern with decontextualised and universalised essence to contextualised and contingent complex wholes. This is where complexity theory seeks the levers of history. The paper posits the notion of inertial momentum as the conceptual link between the principle of emergent phenomena as developed principally in the natural sciences and the notion of socio‐historical change in human society. It is argued that educational and institutional change is less a consequence of effecting change in one particular factor or variable, and more a case of generating momentum in a new direction by attention to as many factors as possible. Complexity theory suggests, in other words, that what it might take to change a school's inertial momentum from an ethos of failure is massive and sustained intervention at every possible level until the phenomenon of learning excellence emerges from this new set of interactions among these new factors, and sustains itself autocatalytically. The paper concludes with a summary consideration of the conditions that contribute to the emergence of new properties and behaviours in a system. (shrink)

Abstract The ‘species problem’ in the philosophy of biology concerns the nature of species. Various solutions have been proposed, including arguments that species are sets, classes, natural kinds, individuals, and homeostatic property clusters. These proposals parallel debates in ecology as to the ontology and metaphysics of populations, communities and ecosystems. A new solution—that species are processes—is proposed and defended, based on Robert Ulanowicz’s metaphysics of process ecology. As with ecological systems, species can be understood as emergent, autocatalytic systems (...) with propensities for centripetality and mutuality in the course of dynamically balancing ascendency (order and persistence) and overhead (randomness and change). The species-as-processes perspective accords with the Ulanowicz’s postulates of process ecology and it can be accommodated by existing theories of species—particularly in a reframing of Richard Boyd’s metaphysics such that species are homeostatic process clusters. Rather than contending that process-based metaphysics is the only, best or true account of species, a pluralist-realist approach is advocated based on the pragmatic principles that are reflected in modern view of species and ecology. If species are understood to be comprised of processes and to be emergent processes themselves, there are important implications for the life sciences, including: animal models in medical and environmental studies, conservation biology, extinction, biodiversity, restoration ecology, and evolutionary biology. Content Type Journal Article Category Invited Paper Pages 1-30 DOI 10.1007/s10516-011-9169-5 Authors Jeffrey A. Lockwood, Department of Philosophy, University of Wyoming, Laramie, WY 82071, USA Journal Axiomathes Online ISSN 1572-8390 Print ISSN 1122-1151. (shrink)

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

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

The paper generalizes Van McGee's well-known result that there are many maximal consistent sets of instances of Tarski's schema to a number of non-classical theories of truth. It is shown that if a non-classical theory rejects some classically valid principle in order to avoid the truth-theoretic paradoxes, then there will be many maximal non-trivial sets of instances of that principle that the non-classical theorist could in principle endorse. On the basis of this it is argued that the idea (...) of classical recapture, which plays such an important role for non-classical logicians, can only be pushed so far. (shrink)

NOWADAYS, even schoolchildren babble about "null sets" and "singletons" and "one-one correspondences," as if they knew what they were talking about. But if they understand even less than their teachers, which seems likely, they must be using the technical jargon with only an illusion of understanding. Beginners are taught that a set having three members is a single thing, wholly constituted by its members but distinct from them. After this, the theological doctrine of the Trinity as "three in one" (...) should be child's play. (shrink)

This article explores the connection between Boolean-valued class models of set theory and the theory of arbitrary objects in roughly Kit Fine’s sense of the word. In particular, it explores the hypothesis that the set-theoretic universe as a whole can be seen as an arbitrary entity. According to this view, the set-theoretic universe can be in many different states. These states are structurally like Boolean-valued models, and they contain sets conceived of as variable or arbitrary objects.

In this paper, we define a new cosine similarity between two interval valued neutrosophic sets based on Bhattacharya’s distance [19]. The notions of interval valued neutrosophic sets (IVNS, for short) will be used as vector representations in 3D-vector space. Based on the comparative analysis of the existing similarity measures for IVNS, we find that our proposed similarity measure is better and more robust. An illustrative example of the pattern recognition shows that the proposed method is simple and effective.

I follow standard mathematical practice and theory to argue that the natural numbers are the finite von Neumann ordinals. I present the reasons standardly given for identifying the natural numbers with the finite von Neumann's (e.g., recursiveness; well-ordering principles; continuity at transfinite limits; minimality; and identification of n with the set of all numbers less than n). I give a detailed mathematical demonstration that 0 is { } and for every natural number n, n is the set of all natural (...) numbers less than n. Natural numbers are sets. They are the finite von Neumann ordinals. (shrink)

REF is the statement that every stationary subset of a cardinal reflects, unless it fails to do so for a trivial reason. The main theorem, presented in Sect. 0, is that under suitable assumptions it is consistent that REF and there is a κ which is κ+n -supercompact. The main concepts defined in Sect. 1 are PT, which is a certain statement about the existence of transversals, and the “bad” stationary set. It is shown that supercompactness (and even the failure (...) of PT) implies the existence of non-reflecting stationary sets. E.g., if REF then for manyλ ⌝ PT(λ, ℵ1). In Sect. 2 it is shown that Easton-support iteration of suitable Levy collapses yield a universe with REF if for every singular λ which is a limit of supercompacts the bad stationary set concentrates on the “right” cofinalities. In Sect. 3 the use of oracle c.c. (and oracle proper—see [Sh-b, Chap. IV] and [Sh 100, Sect. 4]) is adapted to replacing the diamond by the Laver diamond. Using this, a universe as needed in Sect. 2 is forced, where one starts, and ends, with a universe with a proper class of supercompacts. In Sect. 4 bad sets are handled in ZFC. For a regular λ {δ<+ : cfδ<λ} is good. It is proved in ZFC that ifλ=cfλ>ℵ1 then {α<+ : cfα<λ} is the union of λ sets on which there are squares. (shrink)

No semantic theory satisfying certain natural constraints can identify the semantic contents of sentences (the propositions they express), with sets of circumstances in which the sentences are true–no matter how fine-grained the circumstances are taken to be. An objection to the proof is shown to fail by virtue of conflating model-theoretic consequence between sentences with truth-conditional consequence between the semantic contents of sentences. The error underlines the impotence of distinguishing semantics, in the sense of a truth-based theory of logical (...) consequence, and semantics, in the sense of a theory of meaning. (shrink)

We prove a cell decomposition theorem for Presburger sets and introduce a dimension theory for Z-groups with the Presburger structure. Using the cell decomposition theorem we obtain a full classification of Presburger sets up to definable bijection. We also exhibit a tight connection between the definable sets in an arbitrary p-minimal field and Presburger sets in its value group. We give a negative result about expansions of Presburger structures and prove uniform elimination of imaginaries for Presburger (...) structures within the Presburger language. (shrink)

We extend Meyer's 1972 investigation of sets of minimal indices. Blum showed that minimal index sets are immune, and we show that they are also immune against high levels of the arithmetic hierarchy. We give optimal immunity results for sets of minimal indices with respect to the arithmetic hierarchy, and we illustrate with an intuitive example that immunity is not simply a refinement of arithmetic complexity. Of particular note here are the fact that there are three minimal (...) index sets located in Π3 − Σ3 with distinct levels of immunity and that certain immunity properties depend on the choice of underlying acceptable numbering. We show that minimal index sets are never hyperimmune; however, they can be immune against the arithmetic sets. Lastly, we investigate Turing degrees for sets of random strings defined with respect to Bagchi's size-function s. (shrink)

Public reason liberalism is a family of theories according to which liberal political institutions, social structures, and/or basic social rules are politically or morally justified if and only if they can be endorsed from the perspective of each and every free and equal "reasonable and rational" person. Let us call these persons "the members of the justificatory public." Public reason liberalism idealizes the members of the justificatory public in three senses. First, the members of the justificatory public are assumed to (...) be free from at least some of the cognitive distortions and biases that often characterize actual people; we suppose that they generally reason in a sound way on the basis of relevant information. Different versions of public reason liberalism press this idealization quite far, while others insist on a "moderate idealization." Secondly, the members of the justificatory public are idealized insofar as it is assumed that each is a good-willed person, concerned with living with others on terms that are mutually acceptable. In Rawls's language, we suppose that they are not simply rational, but "reasonable": they have a form of "moral sensibility" according to which they are ready to propose fair terms of cooperation, and are willing to abide by them "provided others can be relied on to do likewise." The members of the justificatory public are thus idealized insofar as they are assumed to be moved by their sense of justice, or their aim to respect others as free and equal persons. This leads to a third idealization: by assuming that the members of the justificatory public are moved by this moral sensibility, it is supposed that they do not pay attention to their reasons to ignore their sense of justice . More generally, any specific version of public reason liberalism will hold that some reasons of actual people are not relevant to the justificatory question: the members of the justificatory public "bracket" these reasons in their deliberations. (shrink)

Some reasons to regard the cumulative hierarchy of sets as potential rather than actual are discussed. Motivated by this, a modal set theory is developed which encapsulates this potentialist conception. The resulting theory is equi-interpretable with Zermelo Fraenkel set theory but sheds new light on the set-theoretic paradoxes and the foundations of set theory.

This Paper combines interval- valued neutrouphic sets and rough sets. It studies roughness in interval- valued neutrosophic sets and some of its properties. Finally we propose a Hamming distance between lower and upper approximations of interval valued neutrosophic sets.

We show that every analytic set in the Baire space which is dominating contains the branches of a uniform tree, i.e. a superperfect tree with the property that for every splitnode all the successor splitnodes have the same length. We call this property of analytic sets u-regularity. However, we show that the concept of uniform tree does not suffice to characterize dominating analytic sets in general. We construct a dominating closed set with the property that for no uniform (...) tree whose branches are contained in the closed set, the set of these branches is dominating. We also show that from a Σ1n+1-rapid filter a non-u-regular Π1n-set can be constructed. Finally, we prove that ∑12-Kσ-regularity implies ∑12-u-regularity. (shrink)

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