We study the relative computational power of structures related to the ordered field of reals, specifically using the notion of generic Muchnik reducibility. We show that any expansion of the reals by a continuous function has no more computing power than the reals, answering a question of Igusa, Knight, and Schweber [7]. On the other hand, we show that there is a certain Borel expansion of the reals that is strictly more powerful than the reals and such (...) that any Borel quotient of the reals reduces to it. (shrink)

A recursive enumerator for a function h is an algorithm f which enumerates for an input x finitely many elements including h(x), f is a k(n)-enumerator if for every input x of length n, h(x) is among the first k(n) elements enumerated by f. If there is a k(n)-enumerator for h then h is called k(n)-enumerable. We also consider enumerators which are only A-recursive for some oracle A. We determine exactly how hard it is to enumerate the Kolmogorov function, which (...) assigns to each string x its Kolmogorov complexity: • For every underlying universal machine U, there is a constant a such that C is k(n)-enumerable only if k(n) ≥ n/a for almost all n. • For any given constant k, the Kolmogorov function is k-enumerable relative to an oracle A if and only if A is at least as hard as the halting problem. • There exists an r.e., Turing-incomplete set A such for every non-decreasing and unbounded recursive function k, the Kolmogorov function is k(n)-enumerable relative to A. The last result is obtained by using a relativizable construction for a nonrecursive set A relative to which the prefix-free Kolmogorov complexity differs only by a constant from the unrelativized prefix-free Kolmogorov complexity. Although every 2-enumerator for C is Turing hard for K, we show that reductions must depend on the specific choice of the 2-enumerator and there is no bound on the quantity of their queries. We show our negative results even for strong 2-enumerators as an oracle where the querying machine for any x gets directly an explicit list of all hypotheses of the enumerator for this input. The limitations are very general and we show them for any recursively bounded function g: • For every Turing reduction M and every non-recursive set B, there is a strong 2-enumerator f for g such that M does not Turing reduce B to f. • For every non-recursive set B, there is a strong 2-enumerator f for g such that B is not wtt-reducible to f. Furthermore, we deal with the resource-bounded case and give characterizations for the class ${\rm S}_{2}^{{\rm P}}$ introduced by Canetti and independently Russell and Sundaram and the classes PSPACE, EXP. • ${\rm S}_{2}^{{\rm P}}$ is the class of all sets A for which there is a polynomially bounded function g such that there is a polynomial time tt-reduction which reduces A to every strong 2-enumerator for g. • PSPACE is the class of all sets A for which there is a polynomially bounded function g such that there is a polynomial time Turing reduction which reduces A to every strong 2-enumerator for g. Interestingly, g can be taken to be the Kolmogorov function for the conditional space bounded Kolmogorov complexity. • EXP is the class of all sets A for which there is a polynomially bounded function g and a machine M which witnesses A ∈ PSPACEf for all strong 2-enumerators f for g. Finally, we show that any strong O(log n)-enumerator for the conditional space bounded Kolmogorov function must be PSPACE-hard if P = NP. (shrink)

A mass problem is a set of functions $\omega \to \omega $. For mass problems ${\mathcal {C}}, {\mathcal {D}}$, one says that ${\mathcal {C}}$ is Muchnik reducible to ${\mathcal {D}}$ if each function in ${\mathcal {C}}$ is computed by a function in ${\mathcal {D}}$. In this paper we study some highness properties of Turing oracles, which we view as mass problems. We compare them with respect to Muchnik reducibility and its uniform strengthening, Medvedev reducibility.For $p \in (...) [0,1]$ let ${\mathcal {D}}$ be the mass problem of infinite bit sequences y such that for each computable bit sequence x, the bit sequence $ x {\,\leftrightarrow\,} y$ has asymptotic lower density at most p = y$ ). We show that all members of this family of mass problems parameterized by a real p with $0 p$ for each computable set x. We prove that the Medvedev complexity of the mass problems ${\mathcal {B}}$ is the same for all $p \in $, by showing that they are Medvedev equivalent to the mass problem of functions bounded by ${2^{2}}^{n}$ that are almost everywhere different from each computable function.Next, together with Joseph Miller, we obtain a proper hierarchy of the mass problems of type $\text {IOE}$ : we show that for any order function g there exists a faster growing order function $h $ such that $\text {IOE}$ is strictly above $\text {IOE}$ in the sense of Muchnik reducibility.We study cardinal characteristics in the sense of set theory that are analogous to the highness properties above. For instance, ${\mathfrak {d}} $ is the least size of a set G of bit sequences such that for each bit sequence x there is a bit sequence y in G so that $\underline \rho >p$. We prove within ZFC all the coincidences of cardinal characteristics that are the analogs of the results above. (shrink)

When thinking about the notion of essence or of an essential feature, philosophers typically focus on what I will call the notion of objectual essence. The main aim of this paper is to argue that beside this familiar notion stands another one, the notion of generic essence, which contrary to appearance cannot be understood in terms of the familiar notion, and which also fails to be correctly characterized by certain other accounts which naturally come to mind as well. Some (...) of my objections to these accounts are similar to some of Kit Fine’s compelling objections to the standard modal account of (objectual) essence (Fine 1994). In the light of these objections, Fine advances the view that it is metaphysical necessity which has to be understood in terms of essence, rather than the other way around, and takes essence to be unanalyzable. When formulatinghis view, Fine had only objectual essence in mind (or had both concepts in mind, but assumed that the generic is a special case of the objectual), and for that reason, I will argue, his account fails. I will suggest that Fineans should modify their view, and take it that metaphysical necessity is to be understood in terms of the two notions of essence—a view I myself find appealing. Finally, I will end by suggesting a further move which reduces the objectual to the generic, making metaphysical necessity reducible to generic essence alone—a move with which I myself have some sympathy. (shrink)

We give natural examples of factors of the Muchnik lattice which capture intuitionistic propositional logic , arising from the concepts of lowness, 1-genericity, hyperimmune-freeness and computable traceability. This provides a purely computational semantics for IPC.

In the early nineties of the previous century, leaf languages were introduced as a means for the uniform characterization of many complexity classes, mainly in the range between P and PSPACE . It was shown that the separability of two complexity classes can be reduced to a combinatorial property of the corresponding defining leaf languages. In the present paper, it is shown that every separation obtained in this way holds for every generic oracle in the sense of Blum and (...) Impagliazzo. We obtain several consequences of this result, regarding, e. g., universal oracles, simultaneous separations and type-2 complexity. (shrink)

This paper addresses questions about how the levels of causality (generic and single-case causality) are related. One question is epistemological: can relationships at one level be evidence for relationships at the other level? We present three kinds of answer to this question, categorised according to whether inference is top-down, bottom-up, or the levels are independent. A second question is metaphysical: can relationships at one level be reduced to relationships at the other level? We present three kinds of answer to (...) this second question, categorised according to whether single-case relations are reduced to generic, generic relations are reduced to single-case, or the levels are independent. We then explore causal inference in autopsy. This is an interesting case study, we argue, because it refutes all three epistemologies and all three metaphysics. We close by sketching an account of causality that survives autopsy—the epistemic theory. (shrink)

Let w and M be the countable distributive lattices of Muchnik and Medvedev degrees of non-empty Π1 0 subsets of 2ω, under Muchnik and Medvedev reducibility, respectively. We show that all countable distributive lattices are lattice-embeddable below any non-zero element of w . We show that many countable distributive lattices are lattice-embeddable below any non-zero element of M.

Recently I reported in this journal1 how it became necessary for a judge to settle a dispute between the pharmaceutical industry and certain Dutch pharmacists. It considered the question of whether a pharmacist is permitted, without prior consultation, to give a patient a generic drug instead of the patent drug mentioned on the prescription.Another dispute has now arisen after the pharmaceutical industry discovered that healthcare insurers were paying general practitioners a bonus if they prescribed generic drugs, such as (...) simvastatin or omeprazol (which reduces the …. (shrink)

Antiretroviral pre-exposure prophylaxis (PrEP) is protective against HIV. Low utilization rates amongst HIV vulnerable populations are due in part to the high cost of PrEP. Generic PrEP offers the potential to improve health at significantly reduced costs. In this study, we examine early utilization patterns and prices for generic PrEP. We discuss the opportunities and challenges for generic PrEP to improve health among HIV vulnerable populations.

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

In September 1998 the Dutch Ministry of Health together with the Dutch Society of General Practitioners , the Royal Dutch Society of Pharmacists , and the Dutch Patient and Consumer Federation published a pamphlet entitled: The same medicine in a different coat. Drugs without a trademark, equally effective, but cheaper. Patients could obtain a copy at the local pharmacy or in the waiting room of their general practitioner. It deals with the question whether the name of the patent drug should (...) be written on the prescription or only the active component. This is important because, according to the authors, the costs of health care can be reduced without reducing the quality of the care if the doctor prescribes the generic form. It is also mentioned that another advantage of prescribing the generic form is that, contrary to the …. (shrink)

: The paper presents some mathematical aspects of the question of reducibility of relations. After giving a formal definition of reducibility we present the basic result (due to Herzberger) to the effect that relations of valency at least 3 are always reducible if the cardinality of the relation is at most equal to the cardinality of the underlying set (which is automatically the case if this set is infinite). In contrast to this, if the term "reduction" is given (...) a practicable form, relations on finite sets are "generically" irreducible, as is shown by a simple counting argument. Next we discuss the question of an "intrinsic" criterion for reducibility. Finally we propose a scheme for the graphic representation of reductions. (shrink)

In addition to the familiar cardinal and proportional readings of many and few, there is yet another interpretation, the relative proportional reading. This reading, unlike the ordinary absolute proportional reading, is not conservative. Under the relative reading, 'Many ψs are φs' is true just in case the proportion of φs among ψs is greater than the proportion of φs among members of contextually given alternatives to ψ. I provide a definition of proportional readings that reduces the differences between absolute and (...) relative interpretations to the value of a single parameter.I argue that relative readings are not restricted to many and few, but are also exhibited by the adverbs often and seldom, and by generics. Interpretations of determiners that have been treated as "focus-affected readings," interpretations of adverbs of quantification that have been treated as "pure frequency readings," and interpretations of generics which have been claimed to be cases of "reverse interpretation" or "direct kind predication" are, in fact, instances of relative readings.A: Army reserve service is especially hard on people who own a small business.Q: Do many small-business owners serve in your regiment?A: I don't know what it's like in other regiments –Q: I am not asking you about other regiments.A: I know, but relatively speaking . . .(Interview on Israel Army Radio, June 8th, 2001). (shrink)

The relativized propositional calculus is a system of Boolean formulas with query symbols. A formula in this system is called a one-query formula if the number of occurrences of query symbols is just one. If a one-query formula is a tautology with respect to a given oracle A then it is called a one-query tautology with respect to A. By extending works of Ambos-Spies (1986) and us (2002), we investigate the measure of the class of all oracles A such that (...) the set of all one-query tautologies with respect to A does not p-btt-reduce to A, where p-btt denotes polynomial-time bounded-truth-table. We show that certain Dowd-type generic oracles all belong to the class, and hence measure of the class is one. (shrink)

We study a refined framework of parameterized complexity theory where the parameter dependence of fixed-parameter tractable algorithms is not arbitrary, but restricted by a function in some family . For every family of functions, this yields a notion of -fixed-parameter tractability. If is the class of all polynomially bounded functions, then -fixed-parameter tractability coincides with polynomial time decidability and if is the class of all computable functions, -fixed-parameter tractability coincides with the standard notion of fixed-parameter tractability. There are interesting choices (...) of between these two extremes, for example the class of all singly exponential functions. In this article, we study the general theory of -fixed-parameter tractability. We introduce a generic notion of reduction and two classes -W[P] and -XP, which may be viewed as analogues of NP and EXPTIME, respectively, in the world of -fixed-parameter tractability. (shrink)

While every endpointed interval I in a linear order J is, considered as a linear order in its own right, trivially Muchnik‐reducible to J itself, this fails for Medvedev‐reductions. We construct an extreme example of this: a linear order in which no endpointed interval is Medvedev‐reducible to any other, even allowing parameters, except when the two intervals have finite difference. We also construct a scattered linear order which has many endpointed intervals Medvedev‐incomparable to itself; the only other known construction (...) of such a linear order yields an ordinal of extremely high complexity, whereas this construction produces a low‐level‐arithmetic example. Additionally, the constructions here are “coarse” in the sense that they lift to other uniform reducibility notions in place of Medvedev reducibility itself. (shrink)

A well known fact is that every Lebesgue measurable set is regular, i.e., it includes an F$_{\sigma}$ set of the same measure. We analyze this fact from a metamathematical or foundational standpoint. We study a family of Muchnik degrees corresponding to measure-theoretic regularity at all levels of the effective Borel hierarchy. We prove some new results concerning Nies's notion of LR-reducibility. We build some $\omega$-models of RCA$_0$which are relevant for the reverse mathematics of measure-theoretic regularity.

We study the Lindenbaum algebra ${\fancyscript{A}}$ (WKL o, RCA o) of sentences in the language of second-order arithmetic that imply RCA o and are provable from WKL o. We explore the relationship between ${\Sigma^1_1}$ sentences in ${\fancyscript{A}}$ (WKL o, RCA o) and ${\Pi^0_1}$ classes of subsets of ω. By applying a result of Binns and Simpson (Arch. Math. Logic 43(3), 399–414, 2004) about ${\Pi^0_1}$ classes, we give a specific embedding of the free distributive lattice with countably many generators into ${\fancyscript{A}}$ (...) (WKL o, RCA o). (shrink)

The generic ultrafilter${\cal G}_2 $forced by${\cal P}\left/\left$was recently proved to be neither maximum nor minimum in the Tukey order of ultrafilters, but it was left open where exactly in the Tukey order it lies. We prove${\cal G}_2 $that is in fact Tukey minimal over its projected Ramsey ultrafilter. Furthermore, we prove that for each${\cal G}_2 $, the collection of all nonprincipal ultrafilters Tukey reducible to the generic ultrafilter${\cal G}_k $forced by${\cal P}\left/{\rm{Fin}}^{ \otimes k} $forms a chain of lengthk. (...) Essential to the proof is the extraction of a dense subsetεkfrom +which we prove to be a topological Ramsey space. The spacesεk,k≥ 2, form a hierarchy of high dimensional Ellentuck spaces. New Ramsey-classification theorems for equivalence relations on fronts on εkare proved, extending the Pudlák–Rödl Theorem for fronts on the Ellentuck space, which are applied to find the Tukey and Rudin–Keisler structures below${\cal G}_k $. (shrink)

The study of existentially closed closure algebras begins with Lipparini’s 1982 paper. After presenting new nonelementary axioms for algebraically closed and existentially closed closure algebras and showing that these nonelementary classes are different, this paper shows that the classes of finitely generic and infinitely generic closure algebras are closed under finite products and bounded Boolean powers, extends part of Hausdorff’s theory of reducible sets to existentially closed closure algebras, and shows that finitely generic and infinitely generic (...) closure algebras are elementarily inequivalent. Special properties of algebraically closed, existentially closed, finitely generic, and infinitely generic closure algebras are established along the way. (shrink)

The tower number ${\mathfrak t}$ and the ultrafilter number $\mathfrak {u}$ are cardinal characteristics from set theory. They are based on combinatorial properties of classes of subsets of $\omega $ and the almost inclusion relation $\subseteq ^*$ between such subsets. We consider analogs of these cardinal characteristics in computability theory.We say that a sequence $(G_n)_{n \in {\mathbb N}}$ of computable sets is a tower if $G_0 = {\mathbb N}$, $G_{n+1} \subseteq ^* G_n$, and $G_n\smallsetminus G_{n+1}$ is infinite for each n. (...) A tower is maximal if there is no infinite computable set contained in all $G_n$. A tower ${\left \langle {G_n}\right \rangle }_{n\in \omega }$ is an ultrafilter base if for each computable R, there is n such that $G_n \subseteq ^* R$ or $G_n \subseteq ^* \overline R$ ; this property implies maximality of the tower. A sequence $(G_n)_{n \in {\mathbb N}}$ of sets can be encoded as the “columns” of a set $G\subseteq \mathbb N$. Our analogs of ${\mathfrak t}$ and ${\mathfrak u}$ are the mass problems of sets encoding maximal towers, and of sets encoding towers that are ultrafilter bases, respectively. The relative position of a cardinal characteristic broadly corresponds to the relative computational complexity of the mass problem. We use Medvedev reducibility to formalize relative computational complexity, and thus to compare such mass problems to known ones.We show that the mass problem of ultrafilter bases is equivalent to the mass problem of computing a function that dominates all computable functions, and hence, by Martin’s characterization, it captures highness. On the other hand, the mass problem for maximal towers is below the mass problem of computing a non-low set. We also show that some, but not all, noncomputable low sets compute maximal towers: Every noncomputable (low) c.e. set computes a maximal tower but no 1-generic $\Delta ^0_2$ -set does so.We finally consider the mass problems of maximal almost disjoint, and of maximal independent families. We show that they are Medvedev equivalent to maximal towers, and to ultrafilter bases, respectively. (shrink)

The goal of this paper is to give an account of the pragmatic and social function of slurs, taken as speech acts. I develop a theory of the distinctive illocutionary force and pragmatic structure of slurs. I argue that slurs help to produce subjects who occupy social identities carved out by pernicious ideologies, and that they do this whether or not anyone involved intends for the slur to work that way or has any particular feelings or conscious thoughts associated with (...) using or being targeted by the slur. I offer an Althusser‐inspired account of what an ideology is, as well as of the mechanisms by which slurs can serve to cue and strengthen ideologies. A slur, I argue, is a kind of interpellation, which reduces its target to a generic identity and derogates and subordinates its target. I explore how slurs and ideologies work in tandem to constitute and fortify one another. I end by applying my account to see how well it fits and helps illuminate three quite different, especially complex slurs. (shrink)

Engelhardt’s critiques of “generic chaplaincy” rely on the argument that chaplains are secular; however, professionally certified chaplains must maintain ordination with an ecclesial body. Engelhardt’s concerns are better directed at the academic subfield that supports and trains chaplains: pastoral theology. That field is somewhat guilty of forced ecumenism because it attempts a universal theology rooted in experience and the social sciences rather than the authority of creeds, ecclesial bodies, or traditions. Pastoral theology makes too many sacrifices to the authority (...) of the behavioral sciences without sufficient appreciation for sacramental theology and the spiritual bonds that rituals effect in communities of shared belief. In this model, ritual is reduced to its instrumental value in addressing the emotional needs of patients rather than viewed as efficacious spiritual care. Chaplains should take a holistic approach to theology so that they can better identify the duties and scope of interfaith spiritual care. (shrink)

This dissertation investigates grounding, the relation of non-causal determination whereby one fact obtains in virtue of some other fact or facts. Although considerations of grounding have been central throughout Western philosophy, the last 15-20 years have seen a renaissance of systematic work on grounding in analytic philosophy. The aim of the dissertation is to contribute to our understanding of the nature of grounding and its relation to other central phenomena in metaphysics. -/- Chapter 1 of the dissertation provides a brief (...) presentation of grounding and introduces some distinctions of relevance to later chapters. Chapter 2 discusses the question of whether grounding can be reduced. Four reductive accounts – in terms of fundamentality and supervenience, essence, reduction, and metaphysical laws respectively – are considered. All of these accounts are found to be subject to objections and are consequently rejected. -/- Chapter 3 considers the relation between grounding and another important phenomenon in metaphysics, namely ontological dependence. The question of the relation between the two is approached through an investigation of their modal connection (or lack thereof). It is argued that on the common assumption that grounding is a factive relation, ontological dependence can be shown to be neither necessary nor sufficient for grounding. Furthermore, to get any form of modal connection between the two relations, one needs to move to a non-factive conception of grounding or consider generic ontological dependence and accept a potentially controversial principle about the modal behaviour of grounding. Even then, the modal connection between grounding and ontological dependence is weak and subject to qualifications. -/- Chapter 4 develops and defends the novel notion of metaphysically opaque grounding. It is commonly assumed that grounding is an especially intimate and powerful connection and enables a form of explanation that is particularly strict and illuminating. An arguably related idea is that grounding is necessarily connected with the core features of things – their essences or natures. Metaphysically opaque grounding constitutes a form of grounding that falsifies both of these ideas. Chapter 4 makes the notion of metaphysically opaque grounding precise, motivates the idea that there are genuine cases of such grounding, explores its consequences for the theory of grounding, and defends the notion from potential objections, concluding that there are currently no strong reasons to rule out opaque grounding. (shrink)

In this article, the author argues that the question of educational time is absolutely essential in contemporary debates concerning the fate of the university. In order to examine the nature of educational time, this article first outlines Heidegger's distinction between temporality and Temporality. Second, the author makes a clarification between inauthentic and authentic learning as two forms of educational temporality. Here the article turns to the work of Hubert Dreyfus and Stuart Dreyfus on expert skill building versus standardised or (...) class='Hi'>generic forms of learning. When inauthentic and authentic forms of temporality are brought to light through this distinction, new ways of understanding the convergence and divergence of learning modes open up for critical reflection. Third, the article suggests that while differentiations internal to learning are critical in the struggle to define the nature of education, education cannot be reduced to its temporalising forms. At this point, the work of one of Heidegger's late students, Giorgio Agamben, becomes important for grounding the educational experience in Temporality through study. At stake here is carving out a time in education for enpresencing versus self-projection through action. And finally, the article turns back to Hediegger in order to see the ethical limitations of too quickly collapsing education into learning—even if that learning is authentic. (shrink)

The Probably Approximately Correct (PAC) learning is a machine learning model introduced by Leslie Valiant in 1984. The PACi reducibility refers to the PAC reducibility independent of size and computation time. This reducibility in PAC learning resembles the reducibility in Turing computability. The ordering of concept classes under PAC reducibility is nonlinear, even when restricted to particular concrete examples.Due to the resemblance to Turing Reducibility, we suspected that there could be incomparable PACi and PAC (...) degrees for the PACi and PAC reducibilities as in Turing incomparable degrees. In 1957 Friedberg and in 1956 Muchnik independently solved the Post problem by constructing computably enumerable sets A and B of incomparable degrees using the priority construction method. We adapt this idea to PACi and PAC reducibilities and construct two effective concept classes C and D such that C is not reducible to D and vice versa. When considering PAC reducibility it was necessary to work on the size of an effective concept class, thus we use Kolmogorov complexity to obtain the size. The non-learnability of concept classes in the PAC learning model is explained by the existence of PAC incomparable degrees.Analogous to the Turing jump, we give a jump operation on effective concept classes for the zero jump. To define the zero jump operator for PACi degrees the join of all the effective concept classes is constructed and proved that it is a greatest element. There are many properties proven for existing degrees. Thus we can explore proving those properties to PACi and PAC degrees. But if we prove an embedding from those degrees to PACi and PAC degrees then those properties will be true for PACi and PAC degrees without explicitly proving them.Abstract prepared by Dodamgodage Gihnee M. Senadheera and taken directly from the thesisE-mail: [email protected]: https://www.proquest.com/docview/2717762461/abstract/ACD19F29A8774AF6PQ/1?accountid=13864. (shrink)

In this article, the author argues that the question of educational time is absolutely essential in contemporary debates concerning the fate of the university. In order to examine the nature of educational time, this article first outlines Heidegger's distinction between temporality and Temporality. Second, the author makes a clarification between inauthentic and authentic learning as two forms of educational temporality. Here the article turns to the work of Hubert Dreyfus and Stuart Dreyfus on expert skill building versus standardised or (...) class='Hi'>generic forms of learning. When inauthentic and authentic forms of temporality are brought to light through this distinction, new ways of understanding the convergence and divergence of learning modes open up for critical reflection. Third, the article suggests that while differentiations internal to learning are critical in the struggle to define the nature of education, education cannot be reduced to its temporalising forms. At this point, the work of one of Heidegger's late students, Giorgio Agamben, becomes important for grounding the educational experience in Temporality through study. At stake here is carving out a time in education for enpresencing versus self-projection through action. And finally, the article turns back to Hediegger in order to see the ethical limitations of too quickly collapsing education into learning—even if that learning is authentic. (shrink)

The notion of a [Formula: see text]-generic sequence of P-points is introduced in this paper. It is proved assuming the Continuum Hypothesis that for each [Formula: see text], any [Formula: see text]-generic sequence of P-points can be extended to an [Formula: see text]-generic sequence. This shows that the CH implies that there is a chain of P-points of length [Formula: see text] with respect to both Rudin–Keisler and Tukey reducibility. These results answer an old question of (...) Andreas Blass. (shrink)

Religion in the generic sense is presented as an irreducible mode of human judgment. By emphasizing the generic character of religion Arnett sets himself against the "sectarians," those who would claim unique worth for a particular tradition. By arguing for the irreducible nature of religious judgment he opposes himself to the "secularists," those who would reduce religion to some other mode of judgment, or to a non-cognitive status. The strongest chapters are the third and fourth, which deal with (...) the relation of religion to morality and art, respectively. While vigorous arguments are presented against those who tend to identify religion with one or the other, e.g., Dewey and Santayana, the problems are not oversimplified, and careful attention is given to the affinities as well as the discontinuities involved. The discussion of religion and truth deals more with the generic value of religious judgment than with the question of the truth of particular religious judgments.—M. W. (shrink)

People sometimes try to call others’ beliefs into question by pointing out the contingent causal origins of those beliefs. The significance of such ‘Etiological Challenges’ is a topic that has started attracting attention in epistemology. Current work on this topic aims to show that Etiological Challenges are, at most, only indirectly epistemically significant, insofar as they bring other generic epistemic considerations to the agent’s attention. Against this approach, we argue that Etiological Challenges are epistemically significant in a more direct (...) and more distinctive way. An Etiological Challenge prompts the agent to assess whether her beliefs result from practices of indoctrination, and whether she should reduce confidence in those beliefs, given the anti-reliability of indoctrination as a method of belief-acquisition. Our analysis also draws attention to some of the ways in which epistemic concerns interact with political issues—e.g. relating to epistemic injustice, identity-based discrimination, and segregation—when we’re thinking about the contingent causal origins of our beliefs. (shrink)

Recently a number of complex electrophysiological responses to neurotransmitters have been observed that cannot be described as simple excitation or inhibition. These responses are often characterized as modulatory, although there is no consensus on what defines modulation. Morphological studies reveal certain neurotransmitters stored in what might be release sites without synaptic contact. There is no direct evidence for nonsynaptic release from CNS sites, although such release does occur in the periphery and in invertebrates. Nonsynaptic release might provide a basis for (...) diffuse one-cell-to-many communication, but it might also simply be a means of sending the transmitter to a broader area of a single neuron than occurs in typical synapses. Several kinds of macromolecules have been found to be transported in a retrograde direction – and in some cases transsynaptically. There have been suggestions that some neurons may release more than one type of transmitter. Particularly intriguing is the possibility of release of substances that modulate actions of a primary transmitter. Taken together this range of evidence suggests that neurons may use a variety of forms of molecular communication in addition to traditionally described synaptic transmission.Several authors have suggested modes of communication distinct from classical synaptic transmission and have classified released substances using terms such as neurohumor, neurohormone, neuroregulator, and modulator. These suggestions have the heuristic value of drawing together diverse kinds of data, but it remains to be established that the pieces fit together in that fashion – for example, that complex electrophysiological effects are associated with substances released nonsynaptically. In order to reduce confusion, a flexible, generic approach to nomenclature for substances released from neurons and for hypothetical modes of communication is recommended. Some behavioral implications of nonconventional transmission are considered. (shrink)

Insights from contemporary psychology can illuminate the common psychological processes that facilitate unethical decision making. I will illustrate several of these processes and describe steps that may be taken to reduce or eliminate the undesirable consequences of these processes. A generic problem with these processes is that they are totally invisible to decision makers – i. e., decision makers are convinced that their decisions are ethically and managerially sound.

Why does #RepealedThe8th matter for feminist legal studies? The answers seem obvious in one sense. Feminism has long constituted itself through the struggle for sexual and reproductive justice, and Irish feminism has contributed a significant ‘legal win’ with the landslide vote of approval for lifting abortion restrictions in the referendum on the 25th May 2018. That win comes at a global moment when populist legal engagement is doing significant damage in countries that regard themselves as world leaders, and beyond. #RepealedThe8th (...) offers Ireland, and the world, the actuality that the popular vote, and everything that contributes to it, could be something else. Repeal shows how legal tools like the vote may be made into an expression of care for reproductive lives. This expression is important in recognizing pregnant people as knowing agents who are best placed to decide, and in seeking to do justice to those who contribute to everyday reproductive life. But repeal, like the many who brought it into being, has multiple meanings. #RepealedThe8th matters because it is a moving process of socio-legal translation, which draws on a collective energy, ‘repeal energy’, to turn the travesty that was the Eighth Amendment and all it represents into a search for the rest of reproductive life. In opening up the meaning of the vote, much like feminists elsewhere have opened up the meaning of the strike, Irish feminists have turned public mourning over past mistreatment into a series of reproductive connections. This is not a strategy that can be rolled out. Figuring out #RepealedThe8th will take many tellings. Rather we need to give repeal, and repealers, room to breathe and rest. We need to feel our way through repeal’s production of legal change so that this success is not reduced to some generic transferable set of legal instructions. I begin by reflecting on repeal as a process of feminist socio-legal translation, which shows us how legal change comes about through the motivation of collective joy, the mourning of damaged and lost lives, the sharing of legal knowledge, and the claiming of the rest of reproductive life. (shrink)

Many families refuse to consent to donation from their deceased relatives or over-rule the consent given before death by the patient, but giving families more information about the potential recipients of organs could reduce refusal rates. In this paper, we analyse arguments for and against doing so, and conclude that this strategy should be attempted. While it would be impractical and possibly unethical to give details of actual potential recipients, generic, realistic information about the people who could benefit from (...) organs should be provided to families before they make a decision about donation or attempt to over-rule it. (shrink)

The role of the environment in producing the correct classical limit in the Bohm interpretation of quantum mechanics is investigated, in the context of a model of quantum Brownian motion. One of the effects of the interaction is to produce a rapid approximate diagonalisation of the reduced density matrix in the position representation. This effect is, by itself, insufficient to produce generically quasi-classical behaviour of the Bohmian trajectory. However, it is shown that, if the system particle is initially in an (...) approximate energy eigenstate, then there is a tendency for the Bohmian trajectory to become approximately classical on a longer time-scale. The relationship between this phenomenon and the behaviour of the Wigner function post-decoherence is discussed. (shrink)

This article is on political normativity. It urges scepticism about attempts to reduce political normativity to morality. Modern liberalism leaves a question about how far morality can be accommodated by the form of normativity characteristic of politics. The article casts doubt on whether individual moral norms carry over to collective, for example, political, action, and whether the former ‘trump’ other kinds of reasons in politics. It then sketches an alternative view of politics as an irreducibly collective enterprise. Reasons for acting (...) politically, including the understandings on which perceptions of legitimacy rest, are largely artefacts of the political culture and thus only marginally subject to generic conditions of validity: this is true in particular of liberal acceptability-conditions. Thus legitimacy, though not a redundant notion, must be geared to local political norms. (shrink)

How should we handle ethical issues related to emerging science and technology in a rational way? This is a crucial issue in our time. On the one hand, there is great optimism with respect to technology. On the other, there is pessimism. As both perspectives are based on scarce evidence, they may appear speculative and irrational. Against the pessimistic perspective to emerging technology, it has been forcefully argued that there is a status quo bias (SQB) fuelling irrational attitudes to emergent (...) science and technology and greatly hampering useful development and implementation. Therefore, this article starts by analysing the SQB using human enhancement as a case study. It reveals that SQB may not be as prominent in restricting the implementation of emergent technologies as claimed in the ethics literature, because SQB (a) is fuelled by other and weaker drivers than those addressed in the literature, (b) is at best one amongst many drivers of attitudes towards emergent science and technology, and (c) may not be a particularly prominent driver of irrational decision‐making. While recognizing that SQB can be one driver behind pessimism, this article investigates other and counteracting forces that may be as strong as SQB. Progress bias is suggested as a generic term for the various drivers of unwarranted science and technology optimism. Based on this analysis, a test for avoiding or reducing this progress bias is proposed. Accordingly, we should recognize and avoid a broad range of biases in the assessment of emerging and existing science and technology in order to promote an open and transparent deliberation. (shrink)

The Cyrenaics assert that (1) particular pleasure is the highest good, and happiness is valued not for its own sake, but only for the sake of the particular pleasures that compose it; (2) we should not forego present pleasures for the sake of obtaining greater pleasure in the future. Their anti-eudaimonism and lack of future-concern do not follow from their hedonism. So why do they assert (1) and (2)? After reviewing and criticizing the proposals put forward by Annas, Irwin and (...) Tsouna, I offer two possible reconstructions. In the first reconstruction, I explain claim (1) as follows: happiness has no value above and beyond the value of the particular pleasures that compose it. Also, there is no "structure" to happiness. The Cyrenaics are targeting the thesis that happiness involves having the activities of one's life forming an organized whole, the value of which cannot be reduced to the value of the experiences within that life. I explain claim (2) as follows: a maximally pleasant life is valuable, but the best way to achieve it is to concentrate heedlessly on the present. In the second reconstruction, the good is radically relativized to one's present preferences. The Cyrenaics assert that we desire some particular pleasure, e.g., the pleasure that results from having this drink now. Thus, our telos -- which is based upon our desires -- is this particular pleasure, not (generic) 'pleasure' or the maximization of pleasure over our lifetime. As our desires change, so does our telos. I conclude that the scanty texts we have do not allow us to decide conclusively between these reconstructions, but I give some reasons to support the second over the first. (shrink)

In this essay I argue that the concept of “the body,” ironically generic and a-bodily, is a legacy of the modern political/ecological distinction. I proceed through five sections. First I suggest that the political and the ecological, in spite of a lot of excellent work undermining the nature-culture distinction, remain mutually resistant concepts. In section two I argue that this split can be partially understood through the work of Bruno Latour. For Latour modernity is defined by an attempt to (...) purge culture of nature. This is the “first Great Divide” that constitutes modernity as a concept and in fact a distinct nature-culture. For Latour, this distinction then gets externalized or projected to create a second Great Divide, one imagined by moderns between themselves and other societies. To illustrate the extent of the sway of this distinction, I read a recent and widely influential essay on doing history in the Anthropocene by Dipesh Chakrabarty. Chakrabarty underestimates the role of the nature-culture distinction in the making of climate crisis. Here I also explain that perhaps what is needed is recognition that the human as a morphology both separates itself from ecology and yet—incompatibly—is considered to be an ecological pinnacle. The human isn’t and yet is ecological. Differences among humans, when they are discussed, are attributed unilaterally to the political side of a new political-ecological split. After Latour the nature and culture distinction is questionable, but a political-ecological split survives that attributes the eruptions of discontinuous life entirely to human agency. In the third section I then turn to the work of Frantz Fanon to argue that Latour’s Great Divides are expressed as “Manichaean” in Fanon’s The Wretched of the Earth. The “ecological” import of Fanon’s concept of Manichaean-ism has largely been missed, thanks both to the salience of the political-ecological distinction and to the subtlety of Fanon’s account. But Fanon is not only talking about politics; he’s talking about ecology. The human, Fanon explains, is meant to be a natural master of the planet, a human among animals. The human is master of all “matter” with which all things related to bodies and all bodies related to things are identified. In the fourth section I suggest that it is this modern Manichaean-ism that produces a crucial double bind in the present: Those who fail to be “the body” of modernism are also those most harmed by its elemental (a word I borrow from a critical reading of Maurice Merleau-Ponty and Luce Irigaray for an indistinguishably political and ecological) destruction. Politics, at least in the tradition that gives rise to modern politics, is in fact and has always been the privileging of certain bodies over others. There is no perfectly apolitical bodily feature, though some features of bodies are more politically formative than others. Modernity does nothing to intervene on this. In fact it exacerbates this theme. Neither the discourse of politics nor the discourse of ecology offers refuge to those whose bodies are considered a threat to a human morphology that is planetarily alienated. In the final section of the essay I argue that this partitioning of politics (relations among the humans) from ecology produces “the body” and its generic justice. I return to Latour to argue that he ultimately does not appreciate the double bind discussed in section four. Fanon does. Therefore Fanon offers a better framework for undermining “the political,” not by reducing it to the modern sense of “the ecological,” but by giving attention to what he suggests is the “cortico-visceral” injustice of a modernity that denies its own earthly status. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Kant's DogDavid E. Johnson (bio)In a certain way, it is always too late to pose the question of time.—Jacques Derrida, Margins of PhilosophyIt is well known that Kant was notorious in Königsberg for his strict adherence to routine; he was so regular, Ernst Cassirer reports, that the citizens of Königsberg were able to set their clocks by his movements.1 The most public articulation of this regularity was his daily (...) walk through the city. It is doubtful Kant took a dog along on his constitutional; nevertheless, at the moment in the Critique of Pure Reason that he determines the possibility of the conceptualization of sense perception, which Heidegger considered the very heart of Kant's critical project and which ultimately turns on the regulation of the synthesis of time,2 Kant trots out man's best friend. Although he needs this dog in order to demonstrate the trick of temporal synthesis that makes any sensible conceptuality possible, it is also clear that he needs to keep this dog on a tight leash; he cannot afford to let it run off or go astray. On one reading, then, the Critique of Pure Reason institutes a sort of philosophical leash law. Indeed, Kant holds the dog so tightly that it is always already a dead dog—philosophical roadkill.In literature there is perhaps no more memorable instance of the problem of conceptuality than Jorge Luis Borges's "Funes el memorioso." Within that text, the key moment is the unforgettable example of Ireneo Funes's particular observation of the manifold that others reduce to a dog: "Not only was it difficult for him to see that the generic symbol 'dog' took in all the dissimilar individuals of all shapes and sizes, it irritated him that the 'dog' of three-fourteen in the afternoon, seen in profile, should be indicated by the same noun as the dog at three-fifteen, seen frontally" [Collected Fictions [End Page 19] 136].3 For Funes the inability to synthesize the manifold of experience under a general concept also contaminates the possibility of self-recognition: "His own face in the mirror, his own hands, surprised him every time he saw them" [Collected Fictions 136].4 His surprise at his face and hands as much as his frustration at the generic concept "dog" must be read within the context of Funes's rejection of Locke's suggestion of a rigorously particular language: "In the seventeenth century, Locke postulated (and condemned) an impossible language in which each individual thing—every stone, every bird, every branch—would have its own name; Funes once contemplated a similar language, but discarded the idea as too general, too ambiguous" [Collected Fictions 136].5 In Funes's account, the untenability of Locke's language does not lie in the impossibility of a language grounded upon absolute singularity; he does not argue, for example, that even the most radical empiricism must ultimately depend upon the possibility of ideality. Rather, he argues that insofar as it does not account for the temporal difference of the same from itself, such a language is always already too general.6 In short, "every individual thing, every rock, every bird, every branch," in Funes's eyes, are each and every one more than one. The dog at 3:14 seen in profile is not the same dog when seen at 3:15 in full frontal view. For Funes, the possibility of identity conceived as self-identity and, accordingly, as self-possession over time, is suspended. My face and hands are always different from themselves; they always surprise me; thus they are not mine. In Signs of Borges Sylvia Molloy remarks that Funes's attempt to construct a rigorously particular language can only be "sustained by Funes' attention" and that the words of such a language "finally make sense only to him. Indeed, all that holds them together... is Funes himself" [118, emphasis added]. On Molloy's account, the singularity of Funes's language, grounded as it is only in Funes himself, explains "the narrator's inability to reproduce" it [118]. Yet, Molloy fails to read the maximal effect of Funes's radical empiricism: namely, the... (shrink)

We study the connection between factors of the Medvedev lattice and constructive logic. The algebraic properties of these factors determine logics lying in between intuitionistic propositional logic and the logic of the weak law of the excluded middle (also known as De Morgan, or Jankov, logic). We discuss the relation between the weak law of the excluded middle and the algebraic notion of join-reducibility. Finally we discuss autoreducible degrees.

The dialectic discourse of the ‘gene’ as the unit of heredity deduced from the phenotype, whether an intervening variable or a hypothetical construct, appeared to be settled with the presentation of the molecular model of DNA: the gene was reduced to a sequence of DNA that is transcribed into RNA that is translated into a polypeptide; the polypeptides may fold into proteins that are involved in cellular metabolism and structure, and hence function. This path turned out to be more bewildering (...) the more the regulation of products and functions were uncovered in the contexts of integrated cellular systems. Philosophers struggling to define a unified concept of the gene as the basic entity of genetics confronted those who suggested several different ‘genes’ according to the conceptual frameworks of the experimentalists. Researchers increasingly regarded genes de facto as generic terms for describing their empiric data, and with improved DNA-sequencing capacities these entities were as a rule bottom-up nucleotide sequences that determine functions. Only recently did empiricists return to discuss conceptual considerations, including top-down definitions of units of function that through cellular mechanisms select the DNA sequences which comprise ‘genomic-footprints’ of functional entities. (shrink)

Knowledge about ethical judgments has not advanced appreciably after decades of research. Such research, however, has rarely addressed the possible importance of the content of such judgments; that is, the material appearing in the brief vignettes or scenarios on which survey respondents base their evaluations. Indeed, this content has seemed an afterthought in most investigations. This paper closely examined the vast array of vignettes that have appeared in relevant research in an effort to reduce this proliferation to a more concise (...) set of overarching vignette themes. Six generic themes emerged from this process, labeled here as Dilemma, Classic, Conspiracy, Sophie’s Choice, Runaway Trolley, and Whistle Blowing. Each of these themes is characterized by a unique combination of four key factors that include the extent of protagonist personal benefit from relevant vignette activities and victim salience in vignette descriptions. Theme identification enabled inherent ambiguities in vignettes that threaten construct validity to come into sharp focus, provided clues regarding appropriate vignette construction, and may help to make sense of patterns of empirical findings that heretofore have seemed difficult to explain. (shrink)

Much of the fascination that Petronius' Satyricon holds for its readers originates in the work's gleeful violation of traditional categories of classical genres. Critical terminology makes explicit the issue of unconventionality, as it is reduced to the neutral word ‘work’ in describing the Satyricon, which, as far as we can tell, belongs to no single category , but appropriates elements from many sources in both poetry and prose. Perhaps if we had more evidence with which to compare the work, such (...) as a greater selection of Menippean satire or proto-novels from antiquity, we might be able to identify it more accurately. But the suspicion remains that the intense variety of its evocations, allusions, and parodic passages differentiates it clearly from its component genres without allowing it to settle firmly in any one established genre. A certain amount of ‘Kreuzung der Gattungen’ is, of course, typical of both Alexandrian and consequently Roman texts. But the Satyricon seems to revel in its generic instability; it plays with the notion of ‘literariness’ by revealing impulses from non-literary forms such as mime and subliterary prose fiction, raising this material to an unfamiliar level of literary sophistication even as it debases other traditional genres through parodic techniques. One of the results of this open experimentation with style and decorum is an extremely dense fabric of literary allusion which some would label ‘literary opportunism’. The reader quickly learns to expect intertextual pyrotechnics, swift changes from the sublime to the ridiculous, and humorous incongruities in plot and form, as the stylistic disorder of the text reflects the topsyturvy Petronian world. The modern reader's response to this profusion of referents is to explore the recognizable categories and sources embedded in the work, to tease out the familiar elements in the hope of gaining a better understanding of the whole. Since a great deal of the allusion in the Satyricon functions parodically, there is yet another step necessary in the interpretation, namely taking into account the effect of the decontextualization of language and events from the source material and their recombination and transformation into the new text. (shrink)

In this work we study dimensional theoretical properties of some a±ne dynamical systems. By dimensional theoretical properties we mean Hausdor® dimension and box- counting dimension of invariant sets and ergodic measures on theses sets. Especially we are interested in two problems. First we ask whether the Hausdor® and box- counting dimension of invariant sets coincide. Second we ask whether there exists an ergodic measure of full Hausdor® dimension on these invariant sets. If this is not the case we ask the (...) question, whether at least the variational principle for Haus- dor® dimension holds, which means that there is a sequence of ergodic measures such that their Hausdor® dimension approximates the Hausdor® dimension of the invariant set. It seems to be well accepted by experts that these questions are of great importance in developing a dimension theory of dynamical systems (see the book of Pesin about dimension theory of dynamical systems [PE2]). Dimensional theoretical properties of conformal dynamical systems are fairly well understood today. For example there are general theorems about conformal repellers and hyperbolic sets for conformal di®eomorphisms (see chapter 7 of [PE2]). On the other hand the existence of two di®erent rates of expansion or contraction forces problems that are not captured by a general theory this days. At this stage of de- velopment of the dimension theory of dynamical systems it seems natural to study non conformal examples. This is the ¯rst step to understand the mechanisms that determine dimensional theoretical properties of non conformal dynamical systems. A±ne dynamical systems represent simple examples of non conformal systems. They are easy to de¯ne, but studying their dimensional theoretical properties does never- theless provide challenging mathematical problems and exemplify interesting phe- nomena. We consider here a special class of self-a±ne repellers in dimension two, depending on four parameters (see 2.1.). Furthermore we study a class of attractors of piecewise a±ne maps in dimension three depending on four parameters as well. The last object of our work are projections of these maps that are known as gener- alized Baker's transformations (see 2.2.). The contents of our work is the following: In chapter two we give an overview about some main results in the area of di- mension theory of a±ne dynamical systems and de¯ne the systems we study in this work. We will explain, what is known about the dimensional theoretical properties of these systems and describe what our new results are. In chapter three we then apply symbolic dynamics to our systems. We will introduce explicit shift codings 4 and ¯nd representations of all ergodic measures for our systems using these codings. From chapter four to chapter eight we study dimensional theoretical properties, which our systems generally or generically have. In chapter four we will prove a formula for the box-counting dimension of the repellers and the attractors (see the- orem 4.1.). Then in chapter ¯ve we apply general dimensional theoretical results for ergodic measures found by Ledrappier and Young [LY] and Barreira, Schmeling and Pesin [BPS] to our systems. These results relate the dimension of ergodic measures to metric entropy and Lyapunov exponents. Using this approach we will be able to reduce questions about the dimension of ergodic measures in our context to ques- tions about certain overlapping and especially overlapping self-similar measures on the line. These overlapping self-similar measures are studied in chapter six. Our main theorem extends a result of Peres and Solomyak [PS2] concerning the absolute continuity resp. singularity of symmetric self-similar measures to asymmetric ones (see theorem 6.1.3.). In chapter seven we bring our results together. We prove that we generically (in the sense of Lebesgue measure on a part of the parameter space) have the iden- tity of box-counting and Hausdor® dimension for the repellers and the attractors. (see theorem 7.1.1. and corollary 7.1.2.). This result suggest that one can expect that the identity of box-counting dimension and Hausdor® dimension holds at least generically in some natural classes of non conformal dynamical systems. Furthermore we will see in chapter seven that there generically exists an ergodic measure of full Hausdor® dimension for the repellers. On the other hand the vari- ational principle for Hausdor® dimension is not generic for the attractors. It holds only if we assume a certain symmetry (see theorem 7.1.1.). For generalized Baker's transformations we will ¯nd a part of the parameter space where there generically is an ergodic measure of full dimension and a part where the variational principle for Hausdor® dimension does not hold (see theorem 7.1.3.). Roughly speaking the reason why the variational principle does not hold here is, that if there exists both a stable and an unstable direction one can not generically maximize the dimension in the stable and in the unstable direction at the same time. In an other setting this phenomenon was observed before by Manning and McCluskey [MM]. In chapter eight we extend some results of the last section to invariant sets that correspond to special Markov chains instead of full shifts (see theorem 8.1.1.). In the last two chapters of our work we are interested in number theoretical excep- tions to our generic results. The starting point of our considerations in section nine are results of ErdÄos [ER1] and Alexander and Yorke [AY] that establish singularity and a decrease of dimension for in¯nite convolved Bernoulli measures under special conditions. Using a generalized notion of the Garsia entropy ([GA1/2]) we are able 5 to understand the consequences of number theoretical peculiarities in broader class of overlapping measures (see theorem 9.1.1.). In chapter ten we then analyze number theoretical peculiarities in the context of our dynamical systems. We restrict our attention to a symmetric situation where we generically have the existence of a Bernoulli measure of full dimension and the identity of Hausdor® and box-counting dimension for all of our systems. In the ¯rst section of chapter ten we ¯nd parameter values such that the variational principle for Hausdor® dimension does not hold for the attractors and for the Fat Baker's transformations (see theorem 10.1.1.). These are the ¯rst known examples of dynamical systems for which the variational principle for Hausdor® dimension does not hold because of number theoretical peculiarities of parameter values. For the repellers we have been able to show that under certain number theoretical conditions there is at least no Bernoulli measure of full Hausdor® dimension; the question if the variational principle for Hausdor® dimension holds remains open in this situation. In the second section of chapter ten we will show that the identity for Hausdor® and box-counting dimension can drops because there are number theoretical pecu- liarities. In the context of Weierstrass-like functions this phenomenon was observed by Przytycki and Urbanski [PU]. Our theorem extends this result to a larger class of sets, invariant under dynamical systems (see theorem 10.2.1). At the end of this work the reader will ¯nd two appendices, a list of notations and the list of references. In appendix A we introduce the notions of dimension we use in this work and collect some general facts in dimension theory. In appendix B we state the facts about Pisot-Vijayarghavan number, we need in our analysis of number theoretical peculiarities. The list of notations contains general notations and a table with a summary of notations we use to describe the dynamical systems that we study. Acknowledgments I wish to thank my supervisor JÄorg Schmeling for a lot of valuable discussion and all his help. Also thanks to Luis Barreira for his great hospitality in Lisboa and many interesting comments. This work was done while I was supported by "Promotionstipendium gem. NaFÄoG der Freien UniversitÄat Berlin". (shrink)

In decision making quite often we face permanently changeable and potentially infinite databases when we cannot apply conventional algorithms for choosing a solution. A decision process on infinite databases is called troubleshooting. A decision on these databases is called creative reasoning. One of the first heuristic semi-logical means for creative decision making were proposed in the theory of inventive problem solving by Genrich Altshuller. In this paper, I show that his approach corresponds to the so-called content-generic logic established by (...) Soviet philosophers as an alternative to mathematical logic. The main assumption of content-genetic is that we cannot reduce our thinking to a mathematical combination of signs or to a language as such and our thought is ever cyclic and reflexive so that it contains ever a history. (shrink)

An important technique in large cardinal set theory is that of extending an elementary embedding j: M → N between inner models to an elementary embedding j*: M[G] → N[G*] between generic extensions of them. This technique is crucial both in the study of large cardinal preservation and of internal consistency. In easy cases, such as when forcing to make the GCH hold while preserving a measurable cardinal (via a reverse Easton iteration of α-Cohen forcing for successor cardinals α), (...) the generic G* is simply generated by the image of G. But in difficult cases, such as in Woodin's proof that a hypermeasurable is sufficient to obtain a failure of the GCH at a measurable, a preliminary version of G* must be constructed (possibly in a further generic extension of M[G]) and then modified to provide the required G*. In this article we use perfect trees to reduce some difficult cases to easy ones, using fusion as a substitute for distributivity. We apply our technique to provide a new proof of Woodin's theorem as well as the new result that global domination at inaccessibles (the statement that d (κ) is less than 2κ for inaccessible κ, where d (κ) is the dominating number at κ) is internally consistent, given the existence of 0#. (shrink)

Studies in astrophysical cosmology have served to reveal the incomprehensible fine-tuning of the fundamental constants and cosmological quantities which must obtain if a universe like ours is to be life-permitting. Traditionally, such fine-tuning of the universe for life would have been taken as evidence of divine design. William Dembski’s ’generic chance elimination argument’ provides a framework for evaluating the hypothesis of design with respect to the fine-tuning of the universe. On Dembski’s model the key to a design inference is (...) the elimination of the competing alternatives of physical necessity and chance. In debates over fine-tuning, the former is represented by a ’theory of everything’, which would eliminate or significantly reduce the improbabilities of fine-tuning. The latter takes the shape of the ’many worlds hypothesis’, according to which a ’world ensemble’ of universes exist, thus providing purchase for the ’anthropic principle’. This paper assesses the relative merit. (shrink)

We introduce a new family of jump operators on Borel equivalence relations; specifically, for each countable group [Formula: see text] we introduce the [Formula: see text]-jump. We study the elementary properties of the [Formula: see text]-jumps and compare them with other previously studied jump operators. One of our main results is to establish that for many groups [Formula: see text], the [Formula: see text]-jump is proper in the sense that for any Borel equivalence relation [Formula: see text] the [Formula: see (...) text]-jump of [Formula: see text] is strictly higher than [Formula: see text] in the Borel reducibility hierarchy. On the other hand, there are examples of groups [Formula: see text] for which the [Formula: see text]-jump is not proper. To establish properness, we produce an analysis of Borel equivalence relations induced by continuous actions of the automorphism group of what we denote the full [Formula: see text]-tree, and relate these to iterates of the [Formula: see text]-jump. We also produce several new examples of equivalence relations that arise from applying the [Formula: see text]-jump to classically studied equivalence relations and derive generic ergodicity results related to these. We apply our results to show that the complexity of the isomorphism problem for countable scattered linear orders properly increases with the rank. (shrink)