We define a notion of reducibility for subsets of a second countable T 0 topological space based on relatively continuous relations and admissible representations. This notion of reducibility induces a hierarchy that refines the Baire classes and the Hausdorff–Kuratowski classes of differences. It coincides with Wadge reducibility on zero dimensional spaces. However in virtually every second countable T 0 space, it yields a hierarchy on Borel sets, namely it is well founded and antichains are of length (...) at most 2. It thus differs from the Wadge reducibility in many important cases, for example on the real line $${\mathbb{R}}$$ or the Scott Domain $${\mathcal{P}\omega}$$. (shrink)

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)

We consider Borel sets of finite rank $A \subseteq\Lambda^\omega$ where cardinality of Λ is less than some uncountable regular cardinal K. We obtain a "normal form" of A, by finding a Borel set Ω, such that A and Ω continuously reduce to each other. In more technical terms: we define simple Borel operations which are homomorphic to ordinal sum, to multiplication by a countable ordinal, and to ordinal exponentiation of base K, under the map which sends every Borel set A (...) of finite rank to its Wadge degree. (shrink)

Let A and B be subsets of the space 2 N of sets of natural numbers. A is said to be Wadge reducible to B if there is a continuous map Φ from 2 N into 2 N such that A = Φ -1 (B); A is said to be monotone reducible to B if in addition the map Φ is monotone, that is, $a \subset b$ implies $\Phi (a) \subset \Phi(b)$ . The set A is said to be (...) monotone if a ∈ A and $a \subset b$ imply b ∈ A. For monotone sets, it is shown that, as for Wadge reducibility, sets low in the arithmetical hierarchy are nicely ordered. The ▵ 0 1 sets are all reducible to the (Σ 0 1 but not ▵ 0 1 ) sets, which are in turn all reducible to the strictly ▵ 0 2 sets, which are all in turn reducible to the strictly Σ 0 2 sets. In addition, the nontrivial Σ 0 n sets all have the same degree for n ≤ 2. For Wadge reducibility, these results extend throughout the Borel hierarchy. In contrast, we give two natural strictly Π 0 2 monotone sets which have different monotone degrees. We show that every Σ 0 2 monotone set is actually Σ 0 2 positive. We also consider reducibility for subsets of the space of compact subsets of 2 N . This leads to the result that the finitely iterated Cantor-Bendixson derivative D n is a Borel map of class exactly 2n, which answers a question of Kuratowski. (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)

We show that if Ƒ is any "well-behaved" subset of the Borei functions and we assume the Axiom of Determinacy then the hierarchy of degrees on $P(^\omega \omega )$ induced by Ƒ turns out to look like the Wadge hierarchy (which is the special case where Ƒ is the set of continuous functions).

We present an infinite-game characterization of the well-founded semantics for function-free logic programs with negation. Our game is a simple generalization of the standard game for negation-less logic programs introduced by van Emden [M.H. van Emden, Quantitative deduction and its fixpoint theory, Journal of Logic Programming 3 37–53] in which two players, the Believer and the Doubter, compete by trying to prove a query. The standard game is equivalent to the minimum Herbrand model semantics of logic programming in the sense (...) that a query succeeds in the minimum model semantics iff the Believer has a winning strategy for the game which begins with the Doubter doubting this query. The game for programs with negation that we propose follows the same rules as the standard one, except that the players swap roles every time the play “passes through” negation. We start our investigation by establishing the determinacy of the new game by using some classical tools from the theory of infinite-games. Our determinacy result immediately provides a novel and purely game-theoretic characterization of the semantics of negation in logic programming. We proceed to establish the connections of the game semantics to the existing semantic approaches for logic programming with negation. For this purpose, we first define a refined version of the game that uses degrees of winning and losing for the two players. We then demonstrate that this refined game corresponds exactly to the infinite-valued minimum model semantics of negation [P. Rondogiannis,W.W. Wadge, Minimum model semantics for logic programs with negation-as-failure, ACM Transactions on Computational Logic 6 441–467]. This immediately implies that the unrefined game is equivalent to the well-founded semantics. (shrink)

In [9] we have considered a wide class of "well-behaved" reducibilities for sets of reals. In this paper we continue with the study of Borel reducibilities by proving a dichotomy theorem for the degree-structures induced by good Borel reducibilities. This extends and improves the results of [9] allowing to deal with a larger class of notions of reduction (including, among others, the Baire class ξ functions).

We present the design goals and metatheory of the Monadic Hybrid Calculus, a new formal system that has the same power as the Monadic Predicate Calculus. MHC allows quantification, including relative quantification, in a straightforward way without the use of bound variables, using a simple adaptation of modal logic notation. Thus “all Greeks are mortal” can be written as [G]M. MHC is also ‘hybrid’ in that it has individual constants, which allow us to formulate statements about particular individuals. Thus “Socrates (...) is Athenian and mortal” can be formalised as s.For our proof system, we use a simple adaptation of Beth-style tableaus. The availability of individual constants eliminates the need for labelled deduction.We discuss first the pragmatic and pedagogical advantages of MHC. Then we present the metatheory: formal syntax, semantics, proof rules, soundness, completeness and expressive equivalence with the Monadic Predicate Calculus. (shrink)

Attempts to extend the classical Hausdorff difference hierarchy to the case of partitions of a space to k > 2 subsets lead to non‐equivalent notions. In a hope to identify the “right” extension we consider the extensions appeared in the literature so far: the limit‐, level‐, Boolean and Wadge hierarchies of k ‐partitions. The advantages and disadvantages of the four hierarchies are discussed. The main technical contribution of this paper is a complete characterization of the Wadge degrees of (...) Δ02‐measurable k ‐partitions of the Baire space. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

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

The Axiom of Projective Determinacy implies the existence of a universal $\utilde{\Pi}^{1}_{n}\setminus\utilde{\Delta}^{1}_{n}$ set for every $n \geq 1$. Assuming $\text{\upshape MA}(\aleph_{1})+\aleph_{1}=\aleph_{1}^{\mathbb{L}}$ there exists a universal $\utilde{\Pi}^{1}_{1}\setminus\utilde{\Delta}^{1}_{1}$ set. In ZFC there is a universal $\utilde{\Pi}^{0}_{\alpha}\setminus\utilde{\Delta}^{0}_{\alpha}$ set for every $\alpha$.

Matthias Schröder has asked the question whether there is a weakest discontinuous problem in the topological version of the Weihrauch lattice. Such a problem can be considered as the weakest unsolvable problem. We introduce the discontinuity problem, and we show that it is reducible exactly to the effectively discontinuous problems, defined in a suitable way. However, in which sense this answers Schröder’s question sensitively depends on the axiomatic framework that is chosen, and it is a positive answer if we work (...) in Zermelo–Fraenkel set theory with dependent choice and the axiom of determinacy $\mathsf {AD}$. On the other hand, using the full axiom of choice, one can construct problems which are discontinuous, but not effectively so. Hence, the exact situation at the bottom of the Weihrauch lattice sensitively depends on the axiomatic setting that we choose. We prove our result using a variant of Wadge games for mathematical problems. While the existence of a winning strategy for Player II characterizes continuity of the problem (as already shown by Nobrega and Pauly), the existence of a winning strategy for Player I characterizes effective discontinuity of the problem. By Weihrauch determinacy we understand the condition that every problem is either continuous or effectively discontinuous. This notion of determinacy is a fairly strong notion, as it is not only implied by the axiom of determinacy $\mathsf {AD}$, but it also implies Wadge determinacy. We close with a brief discussion of generalized notions of productivity. (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)

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

We show that, assuming the Axiom of Determinacy, every non-selfdual Wadge class can be constructed by starting with those of level $\omega _1$ and iteratively applying the operations of expansion and separated differences. The proof is essentially due to Louveau, and it yields at the same time a new proof of a theorem of Van Wesep. The exposition is self-contained, except for facts from classical descriptive set theory.

Suppose that any two Π12 sets are comparable in the sense of Wadge degrees. Then every real has a dagger. This argument proceeds by using the Dodd-Jensen core model theory to show that x ε ωω along with, say, “0† implies the existence of a Π12 norm of length u2. As a result of more recent work by John Steel, the same argument will extend to show that the Wadge comparability of all Π12 sets implies Π12 determinacy.

We show that the semi-linear ordering principle for continuous functions implies the determinacy of all Wadge and Lipschitz games.

We prove that the determinacy of all Lipschitz games, the determinacy of all Wadge games, and the semi-linear ordering principle for Lipschitz maps are all equivalent.

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 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 study the behavior of the game operator $$ on Wadge classes of Borel sets. In particular we prove that the classical Moschovakis results still hold in this setting. We also characterize Wadge classes ${\bf{\Gamma }}$ for which the class has the substitution property. An effective variation of these results shows that for all $1 \le \eta < \omega _1^{{\rm{CK}}}$ and $2 \le \xi < \omega _1^{{\rm{CK}}}$, is a Spector class while is not.

Is it possible to give an explicit definition of belief in terms of subjective probability, such that believed propositions are guaranteed to have a sufficiently high probability, and yet it is neither the case that belief is stripped of any of its usual logical properties, nor is it the case that believed propositions are bound to have probability 1? We prove the answer is ‘yes’, and that given some plausible logical postulates on belief that involve a contextual “cautiousness” threshold, there (...) is but one way of determining the extension of the concept of belief that does the job. The qualitative concept of belief is not to be eliminated from scientific or philosophical discourse, rather, by reducing qualitative belief to assignments of resiliently high degrees of belief and a “cautiousness” threshold, qualitative and quantitative belief turn out to be governed by one unified theory that offers the prospects of a huge range of applications. Within that theory, logic and probability theory are not opposed to each other but go hand in hand. (shrink)

There are few more unsettling philosophical questions than this: What happens in attempts to reduce some properties to some other more fundamental properties?

By analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the "reduced coproduct", which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the "ultracoproduct" can be obtained from the usual topological ultraproduct via a compactification process in the style of Wallman and Frink. We prove theorems dealing with (...) the topological structure of reduced coproducts (especially ultracoproducts) and show in addition how one may use this construction to gain information about the category of compact Hausdorff spaces. (shrink)

The class Matr(C) of all matrices for a prepositional logic (, C) is investigated. The paper contains general results with no special reference to particular logics. The main theorem (Th. (5.1)) which gives the algebraic characterization of the class Matr(C) states the following. Assume C to be the consequence operation on a prepositional language induced by a class K of matrices. Let m be a regular cardinal not less than the cardinality of C. Then Matr (C) is the least class (...) of matrices containing K and closed under m-reduced products, submatrices, matrix homomorphisms, and matrix homomorphic counter-images. (shrink)

The universality of critical phenomena is best explained by appeal to the Renormalisation Group (RG). Batterman and Morrison, among others, have claimed that this explanation is irreducible. I argue that the RG account is reducible, but that the higher-level explanation ought not to be eliminated. I demonstrate that the key assumption on which the explanation relies – the scale invariance of critical systems – can be explained in lower-level terms; however, we should not replace the RG explanation with a bottom-up (...) account, rather we should acknowledge that the explanation appeals to dependencies which may be traced down to lower levels. (shrink)

We introduce the computable FS-jump, an analog of the classical Friedman–Stanley jump in the context of equivalence relations on the natural numbers. We prove that the computable FS-jump is proper with respect to computable reducibility. We then study the effect of the computable FS-jump on computably enumerable equivalence relations (ceers).

There are many social psychological theories regarding the nature of prejudice, but only one major theory of prejudice reduction: under the right circumstances, prejudice between groups will be reduced with increased contact. On the one hand, the contact hypothesis has a range of empirical support and has been a major force in social change. On the other hand, there are practical and ethical obstacles to any large-scale controlled test of the hypothesis in which relevant variables can be manipulated. Here we (...) construct a spatialized model that tests the core hypothesis in a large array of game-theoretic agents. Robust results offer a new kind of support for the contact hypothesis: results in simulation do accord with a hypothesis of reduced prejudice with increased contact. The spatialized game-theoretic model also suggests a deeper explanation for at least some of the social psychological phenomena at issue. (shrink)

Mindfulness—the awareness of the present moment and experiences in daily life—contributes to genuine intrinsic and social-oriented values and curbs materialistic and hedonistic values. In the context of materialism, money is power. Avaricious individuals take risks and are likely to engage in dishonesty. Very little research has investigated the effects of mindfulness in reducing the avaricious monetary attitudes and enhancing ethical consumer beliefs. In this study, we theorize that mindfulness improves consumer ethics directly and indirectly by lowering avaricious monetary attitudes. To (...) test our theory, we collected data from 523 individuals with the Mindfulness-Based Stress Reduction training and 307 individuals without MBSR. The results of our whole sample support our theory. Three multiple-group confirmatory factor analyses reveal intriguing discoveries. First, with MBSR training, mindfulness excites consumer ethical beliefs directly and indirectly. Without training, trait mindfulness fails to reduce monetary attitudes—mindfulness training matters. Second, the power of MBSR training holds for participants completing the training within 1 year, but wears off after 1 year—the duration after training matters. Finally, after 1 year, the training retains its strength for those who practice mindfulness, but weakens its power for those who do not—practice matters. We shed light on mindfulness, monetary wisdom, and consumer ethics, in particular, and business ethics, in general. (shrink)

Let equation image. We show that for many reducibilities, the requirement that a relation be intrinsically reducible to the α-th jump of a countable mode A has a syntactic equivalent. Furthermore, we show that many reducibilities coincide in such a situation.

Background Retraction is a mechanism for alerting readers to unreliable material and other problems in the published scientific and scholarly record. Retracted publications generally remain visible and searchable, but the intention of retraction is to mark them as “removed” from the citable record of scholarship. However, in practice, some retracted articles continue to be treated by researchers and the public as valid content as they are often unaware of the retraction. Research over the past decade has identified a number of (...) factors contributing to the unintentional spread of retracted research. The goal of the Reducing the Inadvertent Spread of Retracted Science: Shaping a Research and Implementation Agenda (RISRS) project was to develop an actionable agenda for reducing the inadvertent spread of retracted science. This included identifying how retraction status could be more thoroughly disseminated, and determining what actions are feasible and relevant for particular stakeholders who play a role in the distribution of knowledge. -/- Methods These recommendations were developed as part of a year-long process that included a scoping review of empirical literature and successive rounds of stakeholder consultation, culminating in a three-part online workshop that brought together a diverse body of 65 stakeholders in October–November 2020 to engage in collaborative problem solving and dialogue. Stakeholders held roles such as publishers, editors, researchers, librarians, standards developers, funding program officers, and technologists and worked for institutions such as universities, governmental agencies, funding organizations, publishing houses, libraries, standards organizations, and technology providers. Workshop discussions were seeded by materials derived from stakeholder interviews (N = 47) and short original discussion pieces contributed by stakeholders. The online workshop resulted in a set of recommendations to address the complexities of retracted research throughout the scholarly communications ecosystem. -/- Results The RISRS recommendations are: (1) Develop a systematic cross-industry approach to ensure the public availability of consistent, standardized, interoperable, and timely information about retractions; (2) Recommend a taxonomy of retraction categories/classifications and corresponding retraction metadata that can be adopted by all stakeholders; (3) Develop best practices for coordinating the retraction process to enable timely, fair, unbiased outcomes; and (4) Educate stakeholders about pre- and post-publication stewardship, including retraction and correction of the scholarly record. -/- Conclusions Our stakeholder engagement study led to 4 recommendations to address inadvertent citation of retracted research, and formation of a working group to develop the Communication of Retractions, Removals, and Expressions of Concern (CORREC) Recommended Practice. Further work will be needed to determine how well retractions are currently documented, how retraction of code and datasets impacts related publications, and to identify if retraction metadata (fails to) propagate. Outcomes of all this work should lead to ensuring retracted papers are never cited without awareness of the retraction, and that, in public fora outside of science, retracted papers are not treated as valid scientific outputs. (shrink)

The idea that causation can be reduced to transmission of an amount of some conserved quantity between events is spelled out and defended against important objections. Transmission is understood as a symmetrical relation of copresence in two distinct events. The actual asymmetry of causality has its origin in the asymmetrical character of certain irreversible physical processes and then spreads through the causal net. This conception is compatible with the possibility of backwards causation and with a causal theory of time. Genidentity, (...) the persistence of concrete objects, can be given an explanation in causal terms. The transmission theory is shown to escape difficulties faced by two important alternative theories of causation: Salmon's (1984) Mark Transmission Theory and Dowe's (1992a) Conserved Quantities Theory. (shrink)

This paper presents some of the most promising ways wild animals are currently being helped, as well as other ways of helping that may be implemented easily in the near future. They include measures to save animals affected by harmful weather events, wild animal vaccination programs, and projects aimed at reducing suffering among synanthropic animals. The paper then presents other ways of helping wild animals that, while noncontroversial, may reduce aggregate suffering at the ecosystem level. The paper argues that impracticability (...) and normative objections against reducing wild animal suffering fail against these ways of helping wild animals. (shrink)

I propose that constitution is a case of composition in which, for example, the lump of clay composes the statue. In other words, we can reduce constitution to composition. Composition does all of the work that we want from an account of constitution, and we do not need two separate relations. Along the way, I offer reasons to reject weak supplementation. Acknowledgments (which by my mistake were not included in the journal publication): Many people have given me feedback over the (...) years on this paper or an ancestor of it, and I apologize for oversights in my acknowledgments here. Special thanks to Trenton Merricks for comments on several drafts of this paper. Thanks also for helpful comments and discussions to Eugene Mills, Donald Smith, Peter Sutton, David Liebesman, Charlie Tanksley, Amanda Hicks, Kristl Laux, and to an audience at the University at Buffalo conference in honor of E. J. Lowe, an audience at Amherst College, and an audience at the 2017 APA Eastern Division Meeting. Thank you also to multiple anonymous referees along the way. (shrink)

Although there is much discussion exploring the potential causes of plagiarism, there is limited research available which provides evidence as to the academic interventions which may help reduce this. This paper discusses a bespoke English for Academic Purposes programme introduced at the university level, aimed at improving the academic writing standards of students, reducing plagiarism, and detecting cases of contract cheating. Results from 12 semesters of academic misconduct data demonstrate a 37.01% reduction in instances of detected plagiarism following the intervention, (...) but due to limited data, cannot demonstrate a direct impact on reducing detected rates of contract cheating. The results also show a lower than expected proportion of plagiarised assignments among submissions. (shrink)

The most commonly heard proposals for reducing possible worlds to language succumb to a simple cardinality argument: it can be shown that there are more possible worlds than there are linguistic entities provided by the proposal. In this paper, I show how the standard proposals can be generalized in a natural way so as to make better use of the resources available to them, and thereby circumvent the cardinality argument. Once it is seen just what the limitations are on these (...) more general proposals, it can be clearly seen where the real difficulty lies with any attempt to reduce possible worlds to language. Roughly, the difficulty is this: no actual language could have the descriptive resources needed to represent all the ways things might have been. I conclude by arguing that this same difficulty spells doom for any nominalist or conceptualist proposal for reducing possible worlds. (shrink)

Believing in free will may arise from a biological need for control. People induced to disbelieve in free will show impulsive and antisocial tendencies, suggesting a reduction of the willingness to exert self-control. We investigated whether undermining free will affects two aspects of self-control: intentional inhibition and perceived self-control. We exposed participants either to anti-free will or to neutral messages. The two groups then performed a task that required self-control to inhibit a prepotent response. No-free will participants showed less intentional (...) inhibitions than controls, suggesting a reduction of self-control. We assessed perceived self-control by asking participants whether the response resulted from a deliberate intention or from an impulsive reaction. Perceived self-control was lower in the no-free will group than in control group. Our findings show that undermining free will can degrade self-control and provide insights into how disbelieving in free will leads to antisocial tendencies. (shrink)

This article argues that most of the approaches to the foundations of statistical mechanics have severed their link with the original foundational project, the project of demonstrating how real mechanical systems can behave thermodynamically.

Chemistry and physics are two sciences that are hard to connect. Yet there is significant overlap in their aims, methods, and theoretical approaches. In this book, the reduction of chemistry to physics is defended from the viewpoint of a naturalised Nagelian reduction, which is based on a close reading of Nagel's original text. This naturalised notion of reduction is capable of characterising the inter-theory relationships between theories of chemistry and theories of physics. The reconsideration of reduction also leads to a (...) new characterisation of chemical theories. This book is primarily aimed at philosophers of chemistry and chemists with an interest in philosophy, but is also of interest to the general philosopher of science. (shrink)

A strong reducibility relation between partial numberings is introduced which is such that the reduction function transfers exactly the numbers which are indices under the numbering to be reduced into corresponding indices of the other numbering. The degrees of partial numberings of a given set with respect to this relation form an upper semilattice.In addition, Ershov’s completion construction for total numberings is extended to the partial case: every partially numbered set can be embedded in a set which results from (...) the given set by adding one point and which is enumerated by a total and complete numbering. As is shown, the degrees of complete numberings of the extended set also form an upper semilattice. Moreover, both semilattices are isomorphic.This is not so in the case of the usual, weaker reducibility relation for partial numberings which allows the reduction function to transfer arbitrary numbers into indices. (shrink)

We investigate Borel reducibility between equivalence relations $E(X;p)=X^{\mathbb{N}}/\ell_{p}(X)'s$ where X is a separable Banach space. We show that this reducibility is related to the so called Hölder(α) embeddability between Banach spaces. By using the notions of type and cotype of Banach spaces, we present many results on reducibility and unreducibility between E(L r ; p)'s and E(c 0 ; p)'s for r, p ∈ [1, +∞). We also answer a problem presented by Kanovei in the affirmative by (...) showing that $\mathrm{C}\left({\mathrm{\mathbb{R}}}^{+}\right)/{\mathrm{C}}_{0}\left({\mathrm{\mathbb{R}}}^{+}\ri ght)$ is Borel bireducible to $\mathbb{R}^{\mathbb{N}}/c_{0}$. (shrink)