We investigate some basic descriptive set theory for countably based completely quasi-metrizable topological spaces, which we refer to as quasi-Polish spaces. These spaces naturally generalize much of the classical descriptive set theory of Polish spaces to the non-Hausdorff setting. We show that a subspace of a quasi-Polish space is quasi-Polish if and only if it is Π20 source in the Borel hierarchy. Quasi-Polish spaces can be characterized within the framework (...) of Type-2 Theory of Effectivity as precisely the countably based spaces that have an admissible representation with a Polish domain. They can also be characterized domain theoretically as precisely the spaces that are homeomorphic to the subspace of all non-compact elements of an ω-continuous domain. Every countably based locally compact sober space is quasi-Polish, hence every ω-continuous domain is quasi-Polish. A metrizable space is quasi-Polish if and only if it is Polish. We show that the Borel hierarchy on an uncountable quasi-Polish space does not collapse, and that the Hausdorff–Kuratowski theorem generalizes to all quasi-Polish spaces. (shrink)

The Wadge hierarchy was originally defined and studied only in the Baire space. Here we extend the Wadge hierarchy of Borel sets to arbitrary topological spaces by providing a set-theoretic definition of all its levels. We show that our extension behaves well in second countable spaces and especially in quasi-Polish spaces. In particular, all levels are preserved by continuous open surjections between second countable spaces which implies e.g., several Hausdorff–Kuratowski -type theorems in quasi-Polish spaces. In (...) fact, many results hold not only for the Wadge hierarchy of sets but also for its extension to Borel functions from a space to a countable better quasiorder Q. (shrink)

The wedge hierarchy was originally defined and studied only in the Baire space (and some other zero-dimensional spaces). Here we extend the Wadge hierarchy of Borel sets to arbitrary topological spaces by providing a set-theoretic definition of all its levels. We show that our extension behaves well in second countable spaces and especially in quasi-Polish spaces. In particular, all levels are preserved by continuous open surjections between second countable spaces which implies e.g. several Hausdorff-Kuratowski-type theorems in (...) class='Hi'>quasi-Polish spaces. In fact, many results hold not only for the Wadge hierarchy of sets but also for its extension to Borel functions from a space to a countable better quasiorder Q. (shrink)

We define a quasi-order on Borel functions from a zero-dimensional Polish space into another that both refines the order induced by the Baire hierarchy of functions and generalises the embeddability order on Borel sets. We study the properties of this quasi-order on continuous functions, and we prove that the closed subsets of a zero-dimensional Polish space are well-quasi-ordered by bi-continuous embeddability.

We show that in the Gromov space of isometry classes of pointed proper metric spaces, the equivalence relations defined by existence of coarse quasi-isometries or being at finite Gromov–Hausdorff distance cannot be reduced to the equivalence relation defined by any Polish action.

We prove that the Wadge order on the Borel subsets of the Scott domain is not a well-quasi-order, and that this feature even occurs among the sets of Borel rank at most 2. For this purpose, a specific class of countable 2-colored posets$\mathbb{P}_{emb} $equipped with the order induced by homomorphisms is embedded into the Wadge order on the$\Delta _2^0 $-degrees of the Scott domain. We then show that$\mathbb{P}_{emb} $admits both infinite strictly decreasing chains and infinite antichains with respect to (...) this notion of comparison, which therefore transfers to the Wadge order on the$\Delta _2^0 $-degrees of the Scott domain. (shrink)

If is a Polish metric space of dimension 0, then by Wadge’s lemma, no more than two Borel subsets of X are incomparable with respect to continuous reducibility. In contrast, our main result shows that for any metric space of positive dimension, there are uncountably many Borel subsets of that are pairwise incomparable with respect to continuous reducibility. In general, the reducibility that is given by the collection of continuous functions on a topological space \\) is (...) called the Wadge quasi-order for \\). As an application of the main result, we show that this quasi-order, restricted to the Borel subsets of a Polish space \\), is a well-quasiorder if and only if \\) has dimension 0. Moreover, we give further examples of applications of the construction of graph colorings that is used in the proofs. (shrink)

The Halpern–Läuchli theorem, a combinatorial result about trees, admits an elegant proof due to Harrington using ideas from forcing. In an attempt to distill the combinatorial essence of this proof, we isolate various partition principles about products of perfect Polish spaces. These principles yield straightforward proofs of the Halpern–Läuchli theorem, and the same forcing from Harrington’s proof can force their consistency. We also show that these principles are not ZFC theorems by showing that they put lower bounds on the (...) size of the continuum. (shrink)

Assume that there is no quasi-measurable cardinal not greater than 2ω. We show that for a c. c. c. σ -ideal [MATHEMATICAL DOUBLE-STRUCK CAPITAL I] with a Borel base of subsets of an uncountable Polish space, if [MATHEMATICAL SCRIPT CAPITAL A] is a point-finite family of subsets from [MATHEMATICAL DOUBLE-STRUCK CAPITAL I], then there is a subfamily of [MATHEMATICAL SCRIPT CAPITAL A] whose union is completely nonmeasurable, i.e. its intersection with every non-small Borel set does not belong (...) to the σ -field generated by Borel sets and the ideal [MATHEMATICAL DOUBLE-STRUCK CAPITAL I]. This result is a generalization of the Four Poles Theorem and a result from [3]. (shrink)

This paper is concerned with the proper way to effectivize the notion of a Polish space. A theorem is proved that shows the recursive Polish space structure is not found in the effectively open subsets of a space $${\mathcal {X}}$$ X, and we explore strong evidence that the effective structure is instead captured by the effectively open subsets of the product space $$\mathbb {N}\times {\mathcal {X}}$$ N × X.

We study computable Polish spaces and Polish groups up to homeomorphism. We prove a natural effective analogy of Stone duality, and we also develop an effective definability technique which works up to homeomorphism. As an application, we show that there is a $\Delta ^0_2$ Polish space not homeomorphic to a computable one. We apply our techniques to build, for any computable ordinal $\alpha $, an effectively closed set not homeomorphic to any $0^{}$-computable Polish space; (...) this answers a question of Nies. We also prove analogous results for compact Polish groups and locally path-connected spaces. (shrink)

A Polish space is not always homeomorphic to a computably presented Polish space. In this article, we examine degrees of non-computability of presenting homeomorphic copies of compact Polish spaces. We show that there exists a $\mathbf {0}'$ -computable low $_3$ compact Polish space which is not homeomorphic to a computable one, and that, for any natural number $n\geq 2$, there exists a Polish space $X_n$ such that exactly the high $_{n}$ -degrees (...) are required to present the homeomorphism type of $X_n$. Along the way we investigate the computable aspects of Čech homology groups. We also show that no compact Polish space has a least presentation with respect to Turing reducibility. (shrink)

Louveau and Rosendal [5] have shown that the relation of bi-embeddability for countable graphs as well as for many other natural classes of countable structures is complete under Borel reducibility for analytic equivalence relations. This is in strong contrast to the case of the isomorphism relation, which as an equivalence relation on graphs (or on any class of countable structures consisting of the models of a sentence of L ω ₁ ω ) is far from complete (see [5, 2]). In (...) this article we strengthen the results of [5] by showing that not only does bi-embeddability give rise to analytic equivalence relations which are complete under Borel reducibility, but in fact any analytic equivalence relation is Borel equivalent to such a relation. This result and the techniques introduced answer questions raised in [5] about the comparison between isomorphism and bi-embeddability. Finally, as in [5] our results apply not only to classes of countable structures defined by sentences of ω ₁ ω , but also to discrete metric or ultrametric Polish spaces, compact metrizable topological spaces and separable Banach spaces, with various notions of embeddability appropriate for these classes, as well as to actions of Polish monoids. (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)

We give a partial answer to an important open problem in descriptive set theory, the Decomposability Conjecture for Borel functions on an analytic subset of a Polish space to a separable metrizable space. Our techniques employ deep results from effective descriptive set theory and recursion theory. In fact it is essential to extend several prominent results in recursion theory (e.g. the Shore-Slaman Join Theorem) to the setting of Polish spaces. As a by-product we give both positive (...) and negative results on the Martin Conjecture on the degree preserving Borel functions between Polish spaces. Additionally we prove results about the transfinite version as well as the computable version of the Decomposability Conjecture. (shrink)

We prove the Decomposability Conjecture for functions of Baire class $2$ from a Polish space to a separable metrizable space. This partially answers an important open problem in descriptive set theory.

We investigate separation properties for neighbourhood spaces in some details within a framework of constructive mathematics, and define corresponding separation properties for quasi-apartness spaces. We also deal with separation properties for spaces with inequality.

§ 1. Introduction. In this communication we present some recent results on the classification of Polish metric spaces up to isometry and on the isometry groups of Polish metric spaces. A Polish metric space is a complete separable metric space.Our first goal is to determine the exact complexity of the classification problem of general Polish metric spaces up to isometry. This work was motivated by a paper of Vershik [1998], where he remarks : “The (...) classification of Polish spaces up to isometry is an enormous task. More precisely, this classification is not ‘smooth’ in the modern terminology.” Our Theorem 2.1 below quantifies precisely the enormity of this task.After doing this, we turn to special classes of Polish metric spaces and investigate the classification problems associated with them. Note that these classification problems are in principle no more complicated than the general one above. However, the determination of their exact complexity is not necessarily easier.The investigation of the classification problems naturally leads to some interesting results on the groups of isometries of Polish metric spaces. We shall also present these results below.The rest of this section is devoted to an introduction of some basic ideas of a theory of complexity for classification problems, which will help to put our results in perspective. Detailed expositions of this general theory can be found, e.g., in Hjorth [2000], Kechris [1999], [2001]. (shrink)

We extend the concept of apartness spaces to the concept of quasi-apartness spaces. We show that there is an adjunction between the category of quasi-apartness spaces and the category of neighbourhood spaces, which indicates that quasi-apartness is a more natural concept than apartness. We also show that there is an adjoint equivalence between the category of apartness spaces and the category of Grayson’s separated spaces.

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: [email protected]. (shrink)

We study Polish spaces for which a set of possible distances $A \subseteq R^+$ is fixed in advance. We determine, depending on the properties of A, the complexity of the collection of all Polish metric spaces with distances in A, obtaining also example of sets in some Wadge classes where not many natural examples are known. Moreover we describe the properties that A must have in order that all Polish spaces with distances in that set belong to (...) a given class, such as zero-dimensional, locally compact, etc. These results lead us to give a fairly complete description of the complexity, with respect to Borel reducibility and again depending on the properties of A, of the relations of isometry and isometric embeddability between these Polish spaces. (shrink)

We study the question of how to decompose Hilbert space into a preferred tensor-product factorization without any pre-existing structure other than a Hamiltonian operator, in particular the case of a bipartite decomposition into "system" and "environment." Such a decomposition can be defined by looking for subsystems that exhibit quasi-classical behavior. The correct decomposition is one in which pointer states of the system are relatively robust against environmental monitoring (their entanglement with the environment does not continually and dramatically increase) (...) and remain localized around approximately-classical trajectories. We present an in-principle algorithm for finding such a decomposition by minimizing a combination of entanglement growth and internal spreading of the system. Both of these properties are related to locality in different ways. This formalism could be relevant to the emergence of spacetime from quantum entanglement. (shrink)

We investigate ideals of the form {A⊆ω: Σn∈Axnis unconditionally convergent} where n∈ωis a sequence in a Polish group or in a Banach space. If an ideal onωcan be seen in this form for some sequence inX, then we say that it is representable inX.After numerous examples we show the following theorems: An ideal is representable in a Polish Abelian group iff it is an analytic P-ideal. An ideal is representable in a Banach space iff it is (...) a nonpathological analytic P-ideal.We focus on the family of ideals representable inc0. We characterize this property via the defining sequence of measures. We prove that the trace of the null ideal, Farah’s ideal, and Tsirelson ideals are not representable inc0, and that a tallFσP-ideal is representable inc0iff it is a summable ideal. Also, we provide an example of a peculiar ideal which is representable inℓ1but not in ℝ.Finally, we summarize some open problems of this topic. (shrink)

A quasi-order Q induces two natural quasi-orders on $${\mathcal{P}(Q)}$$, but if Q is a well-quasi-order, then these quasi-orders need not necessarily be well-quasi-orders. Nevertheless, Goubault-Larrecq (Proceedings of the 22nd Annual IEEE Symposium 4 on Logic in Computer Science (LICS’07), pp. 453–462, 2007) showed that moving from a well-quasi-order Q to the quasi-orders on $${\mathcal{P}(Q)}$$ preserves well-quasi-orderedness in a topological sense. Specifically, Goubault-Larrecq proved that the upper topologies of the induced quasi-orders on (...) $${\mathcal{P}(Q)}$$ are Noetherian, which means that they contain no infinite strictly descending sequences of closed sets. We analyze various theorems of the form “if Q is a well-quasi-order then a certain topology on (a subset of) $${\mathcal{P}(Q)}$$ is Noetherian” in the style of reverse mathematics, proving that these theorems are equivalent to ACA 0 over RCA 0. To state these theorems in RCA 0 we introduce a new framework for dealing with second-countable topological spaces. (shrink)

We find the relation between the frequencies received by two observers placed at a given parallel with 180° difference in longitude when they observe a distant light (radio) source. This relation depends on the absolute velocity of the Earth; however, because of the occurrence of aberration, the effect cannot be registered in practice.

We extend a theorem of Barwise and Nadel describing the relationship between approximations of canonical Scott sentences and admissible sets to the case of orbit equivalence relations induced on an arbitrary Polish space by a Polish group action.

If [Formula: see text] is a proper Polish metric space and [Formula: see text] is any countable dense submetric space of [Formula: see text], then the Scott rank of [Formula: see text] in the natural first-order language of metric spaces is countable and in fact at most [Formula: see text], where [Formula: see text] is the Church–Kleene ordinal of [Formula: see text] which is the least ordinal with no presentation on [Formula: see text] computable from [Formula: see (...) text]. If [Formula: see text] is a rigid Polish metric space and [Formula: see text] is any countable dense submetric space, then the Scott rank of [Formula: see text] is countable and in fact less than [Formula: see text]. (shrink)

All spaces are assumed to be separable and metrizable. Consider the following properties of a space X. X is Polish.For every countable crowded Q⊆X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Q \subseteq X}$$\end{document} there exists a crowded Q′⊆Q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${Q'\subseteq Q}$$\end{document} with compact closure.Every closed subspace of X is either scattered or it contains a homeomorphic copy of 2ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${2^\omega}$$\end{document}.Every closed subspace (...) of X is a Baire space. While is the well-known property of being completely Baire, properties and have been recently introduced by Kunen, Medini and Zdomskyy, who named them the Miller property and the Cantor-Bendixson property respectively. It turns out that the implications →→→\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ \rightarrow \rightarrow \rightarrow }$$\end{document} hold for every space X. Furthermore, it follows from a classical result of Hurewicz that all these implications are equivalences if X is coanalytic. Under the axiom of Projective Determinacy, this equivalence result extends to all projective spaces. We will complete the picture by giving a ZFC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sf ZFC}$$\end{document} counterexample and a consistent definable counterexample of lowest possible complexity to the implication ←\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ \leftarrow }$$\end{document} for i=1,2,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${i = 1, 2, 3}$$\end{document}. For one of these counterexamples we will need a classical theorem of Martin and Solovay, of which we give a new proof, based on a result of Baldwin and Beaudoin. Finally, using a method of Fischer and Friedman, we will investigate how changing the value of the continuum affects the definability of these counterexamples. Along the way, we will show that every uncountable completely Baire space has size continuum. (shrink)

We announce two new dichotomy theorems for Borel equivalence relations, and present the results in context by giving an overview of related recent developments.§1. Introduction. For X a Polish space and E a Borel equivalence relation on X, a classification of X up to E-equivalence consists of finding a set of invariants I and a map c : X → I such that xEy ⇔ c = c. To be of any value we would expect I and c (...) to be “explicit” or “definable”. The theory of Borel equivalence relations investigates the nature of possible invariants and provides a hierarchy of notions of classification.The following partial ordering is fundamental in organizing this study. Given equivalence relations E and F on X and Y, resp., we say that E can be Borel reduced to F, in symbolsif there is a Borel map f : X → Y with xEy ⇔ fFf. Then if is an embedding of X/E into Y/F, which is “Borel”.Intuitively, E ≤BF might be interpreted in any one of the following ways: The classi.cation problem for E is simpler than that of F: any invariants for F work as well for E. One can classify E by using as invariants F-equivalence classes. The quotient space X/E has “Borel cardinality” less than or equal to that of Y/F, in the sense that there is a “Borel” embedding of X/E into Y/F. (shrink)

Our aim in this paper is to take quite seriously Heinz Post’s claim that the non-individuality and the indiscernibility of quantum objects should be introduced right at the start, and not made a posteriori by introducing symmetry conditions. Using a different mathematical framework, namely, quasi-set theory, we avoid working within a label-tensor-product-vector-space-formalism, to use Redhead and Teller’s words, and get a more intuitive way of dealing with the formalism of quantum mechanics, although the underlying logic should be modified. (...) We build a vector space with inner product, the Q-space, using the non-classical part of quasi-set theory, to deal with indistinguishable elements. Vectors in Q-space refer only to occupation numbers and permutation operators act as the identity operator on them, reflecting in the formalism the fact of unobservability of permutations. Thus, this paper can be regarded as a tentative to follow and enlarge Heinsenberg’s suggestion that new phenomena require the formation of a new “closed” (that is, axiomatic) theory, coping also with the physical theory’s underlying logic and mathematics. (shrink)

The notion of strong measure zero is studied in the context of Polish groups and general separable metric spaces. An extension of a theorem of Galvin, Mycielski and Solovay is given, whereas the theorem is shown to fail for the Baer–Specker group Zω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb{Z}^{\omega}}}$$\end{document}. The uniformity number of the ideal of strong measure zero subsets of a separable metric space is examined, providing solutions to several problems of Miller and Steprāns (...) :52–59, 2006). (shrink)

Let G be a closed subgroup of S∞ and X be a Polish G -space. To every x ∈ X we associate an admissible set Ax and show how questions about X which involve Baire category can be formalized in Ax.

The issue of abortion has been the topic of heated and frequent debate in post-Communist Poland. Parliamentary debate in 1998—9 centred around a legislative attempt to restrict prenatal testing, specifically amniocentesis, in order to further reduce the numbers of abortions carried out, as it was argued to inevitably result in the termination of pregnancy. Medical professionals are rarely visible as subjects of and authorities on the abortion debate in the Polish context. However, in this debate around prenatal testing, the (...) medical community appear as key commentators and meaning-makers. This article asks the following questions: What role do the medical profession and biomedical knowledge play in the debate around prenatal testing, when abortion is highly politicized? Second, what social meanings and consequences are attributed to prenatal testing? How do these construct the relationship between foetus, pregnant woman and doctor, and what agency and 'rights' are attributed to women in the process of prenatal testing? (shrink)

Let be a separable metrizable space. We establish a criteria for the existence of a metrizable globalization for a given continuous partial action of a separable metrizable group G on. If G and are Polish spaces, we show that the globalization is also a Polish space. We also show the existence of an universal globalization for partial actions of Polish groups.

Utility functions often lack additive separability, presenting an obstacle for decision theoretic axiomatizations. We address this challenge by providing a representation theorem for utility functions of quasi-separable preferences of the form $$u(x,y,z)=f(x,z) + g(y,z)$$ on subsets of topological product spaces. These functions are additively separable only when holding z fixed but are cardinally comparable for different values of z. We then generalize the result to spaces with more than three dimensions and provide applications to belief elicitation, inequity aversion, intertemporal (...) choice, and rank-dependent utility. (shrink)

The generalized Bayes’ rule (GBR) can be used to conduct ‘quasi-Bayesian’ analyses when prior beliefs are represented by imprecise probability models. We describe a procedure for deriving coherent imprecise probability models when the event space consists of a finite set of mutually exclusive and exhaustive events. The procedure is based on Walley’s theory of upper and lower prevision and employs simple linear programming models. We then describe how these models can be updated using Cozman’s linear programming formulation of (...) the GBR. Examples are provided to demonstrate how the GBR can be applied in practice. These examples also illustrate the effects of prior imprecision and prior-data conflict on the precision of the posterior probability distribution. (shrink)

Robust Realists think that there are irreducible, non-natural, and mind-independent moral properties. Quasi-Realists and Relaxed Realists think the same, but interpret these commitments differently. Robust Realists interpret them as metaphysical commitments, to be defended by metaphysical argument. Quasi-Realists and Relaxed Realists say that they can only be interpreted as moral commitments. These theories thus pose a serious threat to Robust Realism, for they apparently undermine the very possibility of articulating the robust metaphysical commitments of this theory. I clarify (...) and respond to this threat, showing that there is in fact space to develop and defend a robust moral ontology. (shrink)

We show that each non-compact Polish group admits a continuous action on a Polish space with non-smooth orbit equivalence relation. We actually construct a free such action. Thus for a Polish group compactness is equivalent to all continuous free actions of this group being smooth. This answers a question of Kechris. We also establish results relating local compactness of the group with its inability to induce orbit equivalence relations not reducible to countable Borel equivalence relations. Generalizing (...) a result of Hjorth, we prove that each non-locally compact, that is, infinite dimensional, separable Banach space has a continuous action on a Polish space with non-Borel orbit equivalence relation, thus showing that this property characterizes non-local compactness among Banach spaces. (shrink)

I seem to know that I won't experience spaceflight but also that if I win the lottery, then I will take a flight into space. Suppose I competently deduce from these propositions that I won't win the lottery. Competent deduction from known premises seems to yield knowledge of the deduced conclusion. So it seems that I know that I won't win the lottery; but it also seems clear that I don't know this, despite the minuscule probability of my winning (...) (if I have a lottery ticket). So we have a puzzle. It seems to generalize, for analogues of the lottery-proposition threaten almost all ordinary knowledge attributions. For example, my apparent knowledge that my bike is parked outside seems threatened by the possibility that it's been stolen since I parked it, a proposition with a low but non-zero probability; and it seems that I don't know this proposition to be false. Familiar solutions to this family of puzzles incur unacceptable costs—either by rejecting deductive closure for knowledge, or by yielding untenable consequences for ordinary attributions of knowledge or of ignorance. After canvassing and criticizing these solutions, I offer a new solution free of these costs. Knowledge that p requires an explanatory link between the fact that p and the belief that p. This necessary but insufficient condition on knowledge distinguishes actual lottery cases from typical, apparently analogous ‘quasi-lottery’ cases. It does yield scepticism about my not winning the lottery and not experiencing spaceflight, but the scepticism doesn't generalize to quasi-lottery cases such as that involving my bike. (shrink)

Strengthening a theorem of Hjorth this paper gives a new characterization of which Polish groups admit compatible complete left invariant metrics. As a corollary it is proved that any Polish group without a complete left invariant metric has a continuous action on a Polish space whose associated orbit equivalence relation is not essentially countable.

Two-classes of phase-space spin quasi-distribution functions are introduced and discussed. The first class of these distributions is based on the delta function construction. It is shown that such a construction can be carried out for an arbitrary spin s and an arbitrary ordering of the spin operators. The second class of the spin distributions is constructed with the help of the spin coherent states. The connection of the spin coherent states to the Stratonovich formalism is established and discussed. (...) It is shown that the c-number phase-space description of quantum fluctuations provides a simple statistical picture of quantum fluctuations of spinoperators in terms of random directions on a unit sphere. For quantum states of the spin system the statistics of these random orientations is given by non-positive spin quasi-distribution functions. It is shown that the application of these spin quasi-distribution functions to the Einstein-Podolsky-Rosen correlations provide an insight into the quantum theory of measurement. (shrink)

This reflection focuses on lived experience with the Technological Other (Quasi-Other) while pursuing creative video and film activities. In the last decade work in the video and film industries has been transformed through digital manipulation and enhancement brought about by increasingly sophisticated computer technologies. The rules of the craft have not changed but the relationship the artist/editor experiences with these new digital tools has brought about increasingly interesting existential experiences in the creative process. How might this new way of (...) being with technology change the craft and the crafter? Through a phenomenological understanding of Martin Heidegger, Maurice Merleau-Ponty, Don Ihde, and their contributions to the human-technology conversation, this essay moves to reveal the lived experiences of artists/editors who use computers to create by means of film and digital video formats. Exploring notions of lived space, lived body, lived time, and lived relation through the computer interface allows for digging deeper into inhabiting technology and experiencing the Technological Other. (shrink)

We introduce a smooth mapping of some discrete space-time symmetries into quasi-continuous ones. Such transformations are related with q-deformations of the dilations of the Euclidean space and with the noncommutative space. We work out two examples of Hamiltonian invariance under such symmetries. The Schrödinger equation for a free particle is investigated in such a noncommutative plane and a connection with anyonic statistics is found.