We prove that the homeomorphism relation between compact spaces can be continuously reduced to the homeomorphism equivalence relation between absolute retracts, which strengthens and simplifies recent results of Chang and Gao, and Cieśla. It follows then that the homeomorphism relation of absolute retracts is Borel bireducible with the universal orbit equivalence relation. We also prove that the homeomorphism relation between regular continua is classifiable by countable structures and hence it is Borel (...) bireducible with the universal orbit equivalence relation of the permutation group on a countable set. On the other hand we prove that the homeomorphism relation between rim-finite metrizable compacta is not classifiable by countable structures. (shrink)

We study the complexity of the classification problem of conjugacy on dynamical systems on some compact metrizable spaces. Especially we prove that the conjugacy equivalence relation of interval dynamical systems is Borel bireducible to isomorphism equivalence relation of countable graphs. This solves a special case of Hjorth’s conjecture which states that every orbit equivalence relation induced by a continuous action of the group of all homeomorphisms of the closed unit interval is classifiable by (...) countable structures. We also prove that conjugacy equivalence relation of Hilbert cube homeomorphisms is Borel bireducible to the universal orbit equivalence relation. (shrink)

LetXbe a standard Borel space, and letEbe acountableBorel equivalence relation onX, i.e., a Borel equivalence relationEfor which every equivalence class [x]Eis countable. By a result of Feldman-Moore [FM],Eis induced by the orbits of a Borel action of a countable groupGonX.The structure of general countable Borel equivalence relations is very little understood. However, a lot is known for the particularly important subclass consisting of hyperfinite relations. A countable Borel equivalence relation is calledhyperfiniteif it is (...) induced by a Borel ℤ-action, i.e., by the orbits of a single Borel automorphism. Such relations are studied and classified in [DJK]. It is shown in Ornstein-Weiss [OW] and Connes-Feldman-Weiss [CFW] that for every Borel equivalence relationEinduced by a Borel action of a countable amenable groupGonXand for every probability measure μ onX, there is a Borel invariant setY⊆Xwith μ = 1 such thatE↾Y is hyperfinite. Motivated by this result, Weiss [W2] raised the question of whether everyEinduced by a Borel action of a countable amenable group is hyperfinite. Later on Weiss showed that this is true forG= ℤn. However, the problem is still open even for abelianG. Our main purpose here is to provide a weaker affirmative answer for general amenableG. We need a definition first. Given two standard Borel spacesX, Y, auniversally measurableisomorphism betweenXandYis a bijection ƒ:X→Ysuch that both ƒ, ƒ-1are universally measurable. Notice now that to assert that a countable Borel equivalence relation onXis hyperfinite is trivially equivalent to saying that there is a standard Borel spaceYand a hyperfinite Borel equivalence relationFonY, which isBorelisomorphic toE, i.e., there is a Borel bijection ƒ:X→YwithxEy⇔ ƒFƒ. We have the following theorem. (shrink)

We prove a number of results motivated by global questions of uniformity in computabi- lity theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of countable groups. We begin by investigating the notion of uniform universality, first proposed by Montalbán, Reimann and Slaman. This notion is a strengthened form of a countable Borel equivalence relation being universal, which we conjecture is equivalent to the usual (...) notion. With this additional uniformity hypothesis, we can answer many questions concerning how countable groups, probability measures, the subset relation, and increasing unions interact with universality. For many natural classes of countable Borel equivalence relations, we can also classify exactly which are uniformly universal. We also show the existence of refinements of Martin’s ultrafilter on Turing invariant Borel sets to the invariant Borel sets of equivalence relations that are much finer than Turing equivalence. For example, we construct such an ultrafilter for the orbit equivalence relation of the shift action of the free group on countably many generators. These ultrafilters imply a number of structural properties for these equivalence relations. (shrink)

We study the complexity of isomorphism of classes of metric structures using methods from infinitary continuous logic. For Borel classes of locally compact structures, we prove that if the equivalence relation of isomorphism is potentially $\mathbf {\Sigma }^0_2$, then it is essentially countable. We also provide an equivalent model-theoretic condition that is easy to check in practice. This theorem is a common generalization of a result of Hjorth about pseudo-connected metric spaces and a result of Hjorth–Kechris about discrete (...) structures. As a different application, we also give a new proof of Kechris’s theorem that orbit equivalence relations of actions of Polish locally compact groups are essentially countable. (shrink)

Let X, Y be Polish spaces,,. We say A is universal for Γ provided that each x‐section of A is in Γ and each element of Γ occurs as an x‐section of A. An equivalence relation generated by a set is denoted by, where. The following results are shown: If A is a set universal for all nonempty closed subsets of Y, then is a equivalence relation and. If A is a set universal (...) for all countable subsets of Y, then is a equivalence relation, and and ; if, then ; if every set is Lebesgue measurable or has the Baire property, then. for, if every set has the Baire property, and E is any equivalence relation, then. (shrink)

We formulate a Borel version of a corollary of Furman's superrigidity theorem for orbit equivalence and present a number of applications to the theory of countable Borel equivalence relations. In particular, we prove that the orbit equivalence relations arising from the natural actions of on the projective planes over the various p-adic fields are pairwise incomparable with respect to Borel reducibility.

We describe punctual categoricity in several natural classes, including binary relational structures and mono-unary functional structures. We prove that every punctually categorical structure in a finite unary language is${\text {PA}}(0')$-categorical, and we show that this upper bound is tight. We also construct an example of a punctually categorical structure whose degree of categoricity is$0''$. We also prove that, with a bit of work, the latter result can be pushed beyond$\Delta ^1_1$, thus showing that punctually categorical structures can possess arbitrarily complex (...) automorphism orbits.As a consequence, it follows that binary relational structures and unary structures are not universal with respect to primitive recursive interpretations; equivalently, in these classes every rich enough interpretation technique must necessarily involve unbounded existential quantification or infinite disjunction. In contrast, it is well-known that both classes are universal for Turing computability. (shrink)

We generalize the notion of analytic/Borel equivalence relations, orbit equivalence relations, and Borel reductions between them to their continuous and quantitative counterparts: analytic/Borel pseudometrics, orbit pseudometrics, and Borel reductions between them. We motivate these concepts on examples and we set some basic general theory. We illustrate the new notion of reduction by showing that the Gromov–Hausdorff distance maintains the same complexity if it is defined on the class of all Polish metric spaces, spaces bounded from below, (...) from above, and from both below and above. Then we show that [Formula: see text] is not reducible to equivalences induced by orbit pseudometrics, generalizing the seminal result of Kechris and Louveau. We answer in negative a question of Ben Yaacov, Doucha, Nies, and Tsankov on whether balls in the Gromov–Hausdorff and Kadets distances are Borel. In appendix, we provide new methods using games showing that the distance-zero classes in certain pseudometrics are Borel, extending the results of Ben Yaacov, Doucha, Nies, and Tsankov. There is a complementary paper of the authors where reductions between the most common pseudometrics from functional analysis and metric geometry are provided. (shrink)

We prove a number of results about countable Borel equivalence relations with forcing constructions and arguments. These results reveal hidden regularity properties of Borel complete sections on certain orbits. As consequences they imply the nonexistence of Borel complete sections with certain features.

We investigate diagonal actions of Polish groups and the related intersection operator on closed subgroups of the acting group. The Borelness of the diagonal orbit equivalence relation is characterized and is shown to be connected with the Borelness of the intersection operator. We also consider relatively tame Polish groups and give a characterization of them in the class of countable products of countable abelian groups. Finally an example of a logic action is considered and its complexity in (...) the Borel reducbility hierarchy determined. (shrink)

This paper develops the foundations of the descriptive set theory of countable Borel equivalence relations on Polish spaces with particular emphasis on the study of hyperfinite, amenable, treeable and universal equivalence relations.

We study the degree structure of the ω-c.e., n-c.e., and Π10 equivalence relations under the computable many-one reducibility. In particular, we investigate for each of these classes of degrees the most basic questions about the structure of the partial order. We prove the existence of the greatest element for the ω-c.e. and n-computably enumerable equivalence relations. We provide computable enumerations of the degrees of ω-c.e., n-c.e., and Π10 equivalence relations. We prove that for all the degree classes (...) considered, upward density holds and downward density fails. (shrink)

We prove Borel superrigidity results for suitably chosen actions of groups of the form SL2, where {p1, …, pt} is a finite nonempty set of primes, and present a number of applications to the theory of countable Borel equivalence relations. In particular, for each prime q, we prove that the orbit equivalence relations arising from the natural actions of SL2 on the projective lines ℚp ∪ {∞}, p ≠ q, over the various p-adic fields are pairwise incomparable (...) with respect to Borel reducibility. (shrink)

Using computable reducibility ⩽ on equivalence relations, we investigate weakly precomplete ceers (a “ceer” is a computably enumerable equivalence relation on the natural numbers), and we compare their class with the more restricted class of precomplete ceers. We show that there are infinitely many isomorphism types of universal (in fact uniformly finitely precomplete) weakly precomplete ceers, that are not precomplete; and there are infinitely many isomorphism types of non‐universal weakly precomplete ceers. Whereas the Visser space (...) of a precomplete ceer always contains an infinite effectively discrete subset, there exist weakly precomplete ceers whose Visser spaces do not contain infinite effectively discrete subsets. As a consequence, contrary to precomplete ceers which always yield partitions into effectively inseparable sets, we show that although weakly precomplete ceers always yield partitions into computably inseparable sets, nevertheless there are weakly precomplete ceers for which no equivalence class is creative. Finally, we show that the index set of the precomplete ceers is Σ3‐complete, whereas the index set of the weakly precomplete ceers is Π3‐complete. As a consequence of these results, we also show that the index sets of the uniformly precomplete ceers and of the e‐complete ceers are Σ3‐complete. (shrink)

We examine how degrees of computably enumerable equivalence relations under computable reduction break down into isomorphism classes. Two ceers are isomorphic if there is a computable permutation of ω which reduces one to the other. As a method of focusing on nontrivial differences in isomorphism classes, we give special attention to weakly precomplete ceers. For any degree, we consider the number of isomorphism types contained in the degree and the number of isomorphism types of weakly precomplete ceers contained in (...) the degree. We show that the number of isomorphism types must be 1 or ω, and it is 1 if and only if the ceer is self-full and has no computable classes. On the other hand, we show that the number of isomorphism types of weakly precomplete ceers contained in the degree can be any member of $[0,\omega ]$. In fact, for any $n \in [0,\omega ]$, there is a degree d and weakly precomplete ceers ${E_1}, \ldots,{E_n}$ in d so that any ceer R in d is isomorphic to ${E_i} \oplus D$ for some $i \le n$ and D a ceer with domain either finite or ω comprised of finitely many computable classes. Thus, up to a trivial equivalence, the degree d splits into exactly n classes.We conclude by answering some lingering open questions from the literature: Gao and Gerdes [11] define the collection of essentially FC ceers to be those which are reducible to a ceer all of whose classes are finite. They show that the index set of essentially FC ceers is ${\rm{\Pi }}_3^0$-hard, though the definition is ${\rm{\Sigma }}_4^0$. We close the gap by showing that the index set is ${\rm{\Sigma }}_4^0$-complete. They also use index sets to show that there is a ceer all of whose classes are computable, but which is not essentially FC, and they ask for an explicit construction, which we provide.Andrews and Sorbi [4] examined strong minimal covers of downwards-closed sets of degrees of ceers. We show that if $\left$ is a uniform c.e. sequence of non universal ceers, then $\left\{ {{ \oplus _{i \le j}}{E_i}|j \in \omega } \right\}$ has infinitely many incomparable strong minimal covers, which we use to answer some open questions from [4].Lastly, we show that there exists an infinite antichain of weakly precomplete ceers. (shrink)

We present some applications of Popa’s Superrigidity Theorem to the theory of countable Borel equivalence relations. In particular, we show that the universal countable Borel equivalence relation E∞ is not essentially free.

In this paper the descriptions of the relation semi-algebra generated by an equivalence element and the relation algebra generated by an equivalence element are unified by functorial constructions. Here the developed techniques and ideas lead to a more manageable conceptual construction of universal objects for the functors of squares and special congruences.

A Polish group G is tame if for any continuous action of G, the corresponding orbit equivalence relation is Borel. When [Formula: see text] for countable abelian [Formula: see text], Solecki [Equivalence relations induced by actions of Polish groups, Trans. Amer. Math. Soc. 347 (1995) 4765–4777] gave a characterization for when G is tame. In [L. Ding and S. Gao, Non-archimedean abelian Polish groups and their actions, Adv. Math. 307 (2017) 312–343], Ding and Gao showed that (...) for such G, the orbit equivalence relation must in fact be potentially [Formula: see text], while conjecturing that the optimal bound could be [Formula: see text]. We show that the optimal bound is [Formula: see text] by constructing an action of such a group G which is not potentially [Formula: see text], and show how to modify the analysis of [L. Ding and S. Gao, Non-archimedean abelian Polish groups and their actions, Adv. Math. 307 (2017) 312–343] to get this slightly better upper bound. It follows, using the results of Hjorth et al. [Borel equivalence relations induced by actions of the symmetric group, Ann. Pure Appl. Logic 92 (1998) 63–112], that this is the optimal bound for the potential complexity of actions of tame abelian product groups. Our lower-bound analysis involves forcing over models of set theory where choice fails for sequences of finite sets. (shrink)

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

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

In the recent philosophical literature, several counterexamples to the interpretative principle that symmetry-related models are physically equivalent have been suggested The Oxford handbook of philosophy of physics, Oxford University Press, Oxford, 2013, Noûs 52:946–981, 2018; Fletcher in Found Phys 50:228–249, 2020). Arguments based on these counterexamples can be understood as arguments from scientific practice of roughly the following form: because in scientific practice such-and-such symmetry-related models are treated as representing distinct physical situations, these models indeed represent distinct physical situations. In (...) this paper, a strategy for analysing arguments of this type is presented and applied to the examples that can be found in the literature. I argue that if we are exclusively interested in models understood as representing entire possible worlds, arguments from scientific practice should involve some additional assumptions to guarantee they are relevant for models understood in this way. However, none of the examples presented in the literature satisfy all these additional assumptions, which leads to the conclusion that arguments from scientific practice based on these examples do not undermine the interpretative principle that different symmetry-related models represent the same possible world. (shrink)

It is shown that the relations of recursive isomorphism on countable trees, groups, Boolean algebras, fields and total orderings are universal countable Borel equivalence relations, thus providing a countable analogue of the Borel completeness of the isomorphism relations on these same classes. I.

Top-down causation is often taken to be a metaphysically suspicious type of causation that is found in a few complex systems, such as in human mind-body relations. However, as Ellis and others have shown, top-down causation is ubiquitous in physics as well as in biology. Top-down causation occurs whenever specific dynamic behaviors are realized or selected among a broader set of possible lower-level states. Thus understood, the occurrence of dynamic and structural patterns in physical and biological systems presents a problem (...) for reductionist positions. We illustrate with examples of universality and functional equivalence classes how higher-level behaviors can be multiple realized by distinct lower-level systems or states. Multiple realizability in both contexts entails what Ellis calls “causal slack” between levels, or what others understand as relative explanatory autonomy. To clarify these notions further, we examine procedures for upscaling in multi-scale modeling. We argue that simple averaging strategies for upscaling only work for simplistic homogenous systems, because of the scale-dependency of characteristic behaviors in multi-scale systems. We suggest that this interpretation has implications for what Ellis calls mechanical top-down causation, as it presents a stronger challenge to reductionism than typically assumed. (shrink)

Carl F. H. Henry serves as a fruitful resource for the integration of faith and learning. The central issue in Christian scholarship is to properly associate the revelation of God with the knowledge of God’s world across all academic disciplines. The particular effort of this article is to demonstrate the clarity Henry provides as it relates to general revelation, special revelation, and knowledge explored in a comprehensive university setting. Building on Henry’s clarity, an orientation of knowledge to Jesus Christ, a (...) proposal for the resulting vision for Christian scholarship, and habits of Christian educational institutions follows. (shrink)

We live in a world in which we are socially, politically, economically, and environmentally connected with other people. Online communication has facilitated people coming together from different parts of the world. In terms of social justice movements, people have come together to share ideas about how they perceive social inequality and how to address it, which is what academics call critical consciousness. While scholars have explored critical consciousness in the American context, whether it operates on a global scale is under-explored. (...) To address this question, we administered the Critical Consciousness Scale (a validated survey) with students from the United States, Iran, and Ukraine. Our findings demonstrate that critical consciousness maintains its factor structure across the entire sample, meaning that students from these three countries share some notions of critical consciousness. However, when comparing national groups, we find that critical consciousness is defined differently by students in different countries. In a practical sense, these findings mean that some aspects of critical consciousness are shared, but there are important differences in how it is perceived and how its components relate to one another. By attempting to understand critical consciousness internationally, this study serves as a cautionary narrative for international solidarity movements organized around the goal of social justice. (shrink)

In this paper, we explore some of the consequences of Martin’s Conjecture on degree invariant Borel maps. These include the strongest conceivable ergodicity result for the Turing equivalence relation with respect to the filter on the degrees generated by the cones, as well as the statement that the complexity of a weakly universal countable Borel equivalence relation always concentrates on a null set.

The concept of polarity is pervasive in natural language. It relates syntax, semantics and pragmatics narrowly, Semantics: an international handbook of natural language meaning, De Gruyter Mouton, Berlin, 2011; Israel in The grammar of polarity: pragmatics, sensitivity, and the logic of scales, Cambridge studies in linguistics, Cambridge University Press, Cambridge, 2014), it refers to items of many syntactic categories such as nouns, verbs and adverbs. Neutral polarity items appear in affirmative and negative sentences, negative polarity items cannot appear in affirmative (...) sentences, and positive polarity items cannot appear in negative sentences. A way of reasoning in Natural Language is through Natural Logic. This logic is based on the concept of polarity in order to make the meaning of a sentence weaker o stronger without changing its truth value. There exist many proposals to compute polarity in the Natural Logic context, the most widely known are the ones by: van Benthem, Sánchez-Valencia, Dowty, and van Eijck, 6th international Tbilisi symposium on logic, language, and computation, Batumi, Georgia, Springer, 2007). If Natural Logic is going to be used, as an inferential mechanism between text fragments, in Natural Language Processing applications such as text summarization, question answering, and information extraction, it is a priority to know what the existing relationship among the aforementioned algorithms is; for example, to implement the most general. We show in this paper the equivalence among the analyzed algorithms, filling a gap in Natural Logic research, particularly in computing polarity, and the soundness of their algorithms. (shrink)

We compute here the Borel complexity of the relation of isometry between separable Banach spaces, using results of Gao, Kechris [2], Mayer-Wolf [5], and Weaver [8]. We show that this relation is Borel bireducible to the universal relation for Borel actions of Polish groups. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim).

Kantians are increasingly deserting the universal law formula in favor of the humanity formula. The former, they argue, is open to various decisive objections; the two are not equivalent; and it is only by appealing to the humanity formula that Kant can reliably generate substantive implications from his theory of an acceptable sort. These assessments of the universal law formula, which clash starkly with Kant's own assessment of it, are based on various widely accepted interpretative assumptions. These assumptions, (...) it is argued in this article, depend on misleading translations of key terms; selective attention to Kant's concrete examples; not taking seriously Kant's theoretical claims about the relations among his various ideas; and a failure to take into account Kant's idiosyncratic definitions of key concepts. The article seeks to right these interpretative wrongs, and finds that the universal law formula is not open to many of the standard objections. (shrink)

Necessitism, Contingentism, and Theory Equivalence is a dissertation on issues in higher-order modal metaphysics. Consider a modal higher-order language with identity in which the universal quantifier is interpreted as expressing universal quantification and the necessity operator is interpreted as expressing metaphysical necessity. The main question addressed in the dissertation concerns the correct theory formulated in this language. A different question that also takes centre stage in the dissertation is what it takes for theories to be equivalent.The whole (...) dissertation consists of an extended argument in defence of the truth of two seemingly inconsistent higher-order modal theories, specifically: 1.Plantingan Moderate Contingentism, a theory based on Plantinga’s [1] modal metaphysics that is committed to, among other things, the contingent being of some individuals and the necessary being of all possible higher-order entities;2.Williamsonian Thorough Necessitism, a theory advocated by Williamson [3] which is committed to, among other things, the necessary being of every possible individual as well as of every possible higher-order entity.Part of the case for these theories’ joint truth relies on defences of the following metaphysical theses: Thorough Serious Actualism, the thesis that no things could have been related while being nothing, and Higher-Order Necessitism, the thesis that necessarily, every higher-order entity is necessarily something. It is shown that Thorough Serious Actualism and Higher-Order Necessitism are both implicit commitments of very weak logical theories. The defence of Higher-Order Necessitism constitutes a powerful challenge to Stalnaker’s [2] Thorough Contingentism, a theory committed to, among other things, the view that there could have been some individuals as well as some entities of any higher-order that could have been nothing.In the dissertation it is argued that Plantingan Moderate Contingentism and Williamsonian Thorough Necessitism are in fact equivalent, even if they appear to be jointly inconsistent. The case for this claim relies on the Synonymy account, a novel account of theory equivalence developed and defended in the dissertation. According to this account, theories are equivalent just in case they have the same commitments and conception of logical space.By way of defending the Synonymy account’s adequacy, the account is applied to the debate between noneists, proponents of the view that some things do not exist, and Quineans, proponents of the view that to exist just is to be some thing. The Synonymy account is shown to afford a more nuanced and better understanding of that debate by revealing that what noneists and Quineans are really disagreeing about is what expressive resources are available to appropriately describe the world.By coupling a metatheoretical result with tools from the philosophy of language, it is argued that Plantingan Moderate Contingentism and Williamsonian Thorough Necessitism are synonymous theories, and so, by the lights of the Synonymy account, equivalent. Given the defence of their extant commitments made in the dissertation, it is concluded that Plantingan Moderate Contingentism and Williamsonian Thorough Necessitism are both correct. A corollary of this result is that the dispute between Plantingans and Williamsonians is, in an important sense, merely verbal. For if two theories are equivalent, then they “require the same of the world for their truth.”Thus, the results of the dissertation reveal that if one speaks as a Plantingan while advocating Plantingan Moderate Contingentism, or as a Williamsonian while advocating Williamsonian Thorough Necessitism, then one will not go wrong. Notwithstanding, one will still go wrong if one speaks as a Plantingan while advocating Williamsonian Thorough Necessitism, or as a Williamsonian while advocating Plantingan Moderate Contingentism.On the basis of a conception of the individual constants and predicates of second-order modal languages as strongly Millian, i.e., as having actually existing entities as their semantic values, in the appendix are presented second-order modal logics consistent with Stalnaker’s Thorough Contingentism. Furthermore, it is shown there that these logics are strong enough for applications of higher-order modal logic in mathematics, a result that constitutes a reply to an argument to the contrary by Williamson [3]. Finally, these logics are proven to be complete relative to particular “thoroughly contingentist” classes of models. (shrink)

We show that the enveloping space \ of a partial action of a Polish group G on a Polish space \ is a standard Borel space, that is to say, there is a topology \ on \ such that \\) is Polish and the quotient Borel structure on \ is equal to \\). To prove this result we show a generalization of a theorem of Burgess about Borel selectors for the orbit equivalence relation induced by a group (...) action and also show that some properties of the Vaught’s transform are valid for partial actions of groups. (shrink)

In this paper we propose and defend the _Synonymy account_, a novel account of metaphysical equivalence which draws on the idea (Rayo in _The Construction of Logical Space_, Oxford University Press, Oxford, 2013) that part of what it is to formulate a theory is to lay down a theoretical hypothesis concerning logical space. Roughly, two theories are synonymous—and so, in our view, equivalent—just in case (i) they take the same propositions to stand in the same entailment relations, and (ii) (...) they are committed to the truth of the same propositions. Furthermore, we put our proposal to work by showing that it affords a better and more nuanced understanding of the debate between Quineans and noneists. Finally we show how the _Synonymy account_ fares better than some of its competitors, specifically, McSweeney’s (Philosophical Perspectives 30(1):270–293, 2016) epistemic account and Miller’s (Philosophical Quarterly 67(269):772–793, 2017) hyperintensional account. (shrink)

Defunct satellites and other technological waste are increasingly occupying Earth’s orbital space, a region designated as one of the global commons. These dilapidated technologies that were commissioned to sustain the production and exchange of data, information, and images are an extraterrestrial equivalent of the media devices which are discarded on Earth. While indicating the extension of technological momentum in the shared commons of space, orbital debris conveys the dark side of media materialities beyond the globe. Its presence and movements interfere (...) with a gamut of governmental, commercial, and scientific operations, contesting the strategies of its management and control and introducing orbital uncertainty and disorder in the global affairs of law, politics, economics, and techno-science. I suggest that this debris formation itself functions as media apparatus —it not only embodies but also exerts its own effects upon the material and social relations that structure our ways of life, perplexing dichotomies between the common and owned, governed and ungovernable, wealth and waste. I explore these effects of debris, framing its situation in the orbital commons as a vital matter of concern for studies of the human relationship with media technologies and their waste. (shrink)

Hjorth introduced a Scott analysis for general Polish group actions, and asked whether his notion of rank satisfies a boundedness principle similar to the one of Scott rank—namely, if the orbit equivalence relation is Borel, then Hjorth ranks are bounded. We answer Hjorth’s question positively. As a corollary we prove the following conjecture of Hjorth—for every limit ordinal \, the set of elements whose orbit is of complexity less than \ is a Borel set.

The study of continuous group actions is ubiquitous in mathematics, and perhaps the most general kinds of actions for which we can hope to prove theorems in just ZFC are those where a Polish group acts on a Polish space.For this general class we can find works such as [29] that build on ideas from ergodic theory and examine actions of locally compact groups in both the measure theoretic and topological contexts. On the other hand a text in model theory, (...) such as [12], will typically consider issues bearing on the actions by the symmetric group of all permutations of the integers. More generally [1] shows that the orbit equivalence relations induced by closed subgroups of the infinite symmetric group can be reduced to the isomorphism relation on corresponding classes of countable models.This paper considers a third category formed by the continuous actions of separable Banach spaces on Polish spaces. These examples cannot be subsumed under the two earlier headings, and it is known from [10] that theBorel cardinalitiesof the quotient spaces that arise from such actions are incomparable with the equivalence relations induced by the symmetric group or any locally compact Polish group action.One of the first things to be addressed concerns the complexity of these equivalence relations. This question forappears in [1]. (shrink)

We study effectively inseparable (abbreviated as e.i.) prelattices (i.e., structures of the form$L = \langle \omega, \wedge, \vee,0,1,{ \le _L}\rangle$whereωdenotes the set of natural numbers and the following four conditions hold: (1)$\wedge, \vee$are binary computable operations; (2)${ \le _L}$is a computably enumerable preordering relation, with$0{ \le _L}x{ \le _L}1$for everyx; (3) the equivalence relation${ \equiv _L}$originated by${ \le _L}$is a congruence onLsuch that the corresponding quotient structure is a nontrivial bounded lattice; (4) the${ \equiv _L}$-equivalence classes (...) of 0 and 1 form an effectively inseparable pair of sets). Solving a problem in (Montagna & Sorbi, 1985) we show (Theorem 4.2), that ifLis an e.i. prelattice then${ \le _L}$is universal with respect to all c.e. preordering relations, i.e., for every c.e. preordering relationRthere exists a computable functionfreducingRto${ \le _L}$, i.e.,$xRy$if and only if$f\left( x \right){ \le _L}f\left( y \right)$, for all$x,y$. In fact (Corollary 5.3)${ \le _L}$is locally universal, i.e., for every pair$a{ < _L}b$and every c.e. preordering relationRone can find a reducing functionffromRto${ \le _L}$such that the range offis contained in the interval$\left\{ {x:a{ \le _L}x{ \le _L}b} \right\}$. Also (Theorem 5.7)${ \le _L}$is uniformly dense, i.e., there exists a computable functionfsuch that for every$a,b$if$a{ < _L}b$then$a{ < _L}f\left( {a,b} \right){ < _L}b$, and if$a{ \equiv _L}a\prime$and$b{ \equiv _L}b\prime$then$f\left( {a,b} \right){ \equiv _L}f\left( {a\prime,b\prime } \right)$. Some consequences and applications of these results are discussed: in particular (Corollary 7.2) for$n \ge 1$the c.e. preordering relation on${{\rm{\Sigma }}_n}$sentences yielded by the relation of provable implication of any c.e. consistent extension of Robinson’s systemRorQis locally universal and uniformly dense; and (Corollary 7.3) the c.e. preordering relation yielded by provable implication of any c.e. consistent extension of Heyting Arithmetic is locally universal and uniformly dense. (shrink)

In the presence of large cardinals, or sufficient determinacy, every equivalence relation in L(R) either admits a wellordered separating family or continuously reduces E 0.

We establish an Easton theorem for the least supercompact cardinal that is consistent with the level by level equivalence between strong compactness and supercompactness. In both our ground model and the model witnessing the conclusions of our theorem, there are no restrictions on the structure of the class of supercompact cardinals. We also briefly indicate how our methods of proof yield an Easton theorem that is consistent with the level by level equivalence between strong compactness and supercompactness in (...) a universe with a restricted number of large cardinals. We conclude by posing some related open questions. (shrink)

It is known that every non-universal self-full degree in the structure of the degrees of computably enumerable equivalence relations (ceers) under computable reducibility has exactly one strong minimal cover. This leaves little room for embedding wide partial orders as initial segments using self-full degrees. We show that considerably more can be done by staying entirely inside the collection of non-self-full degrees. We show that the poset can be embedded as an initial segment of the degrees of ceers with (...) infinitely many classes. A further refinement of the proof shows that one can also embed the free distributive lattice generated by the lower semilattice as an initial segment of the degrees of ceers with infinitely many classes. (shrink)

Abstract.As an undergraduate at Cambridge, Newton entered into his ‘Waste Book’ an assumption that we have named the Equivalence Assumption (The Younger): ‘‘ If a body move progressively in some crooked line [about a center of motion]..., [then this] crooked line may bee conceived to consist of an infinite number of streight lines. Or else in any point of the croked line the motion may bee conceived to be on in the tangent.’’ In this assumption, Newton somewhat imprecisely describes (...) two mathematical models, a ‘‘polygonal limit model’’ and a ‘‘tangent deflected model,’’ for ‘‘one-body motion,’’ that is, for the motion of a ‘‘body in orbit about a fixed center,’’ and then claims that these two models are equivalent. In the first part of this paper, we study the Principia to determine how the elder Newton would more carefully describe the polygonal limit and tangent deflected models. From these more careful descriptions, we then create Equivalence Assumption (The Elder), a precise interpretation of Equivalence Assumption (The Younger) as it might have been restated by Newton, after say 1687. We then review certain portions of the Waste Book and the Principia to make the case that, although Newton never restates nor even alludes to the Equivalence Assumption after his youthful Waste Book entry, still the polygonal limit and tangent deflected models, as well as an unspoken belief in their equivalence, infuse Newton’s work on orbital motion. In particular, we show that the persuasiveness of the argument for the Area Property in Proposition 1 of the Principia depends crucially on the validity of Equivalence Assumption (The Elder). After this case is made, we present the mathematical analysis required to establish the validity of the Equivalence Assumption (The Elder). Finally, to illustrate the fundamental nature of the resulting theorem, the Equivalence Theorem as we call it, we present three significant applications: we use the Equivalence Theorem first to clarify and resolve questions related to Leibniz’s ‘‘polygonal model’’ of one-body motion; then to repair Newton’s argument for the Area Property in Proposition 1; and finally to clarify and resolve questions related to the transition from impulsive to continuous forces in ‘‘De motu’’ and the Principia. (shrink)

A sentence like every circle is blue might be understood in terms of individuals and their properties or in terms of a relation between groups. Relatedly, theorists can specify the contents of universally quantified sentences in first-order or second-order terms. We offer new evidence that this logical first-order vs. second-order distinction corresponds to a psychologically robust individual vs. group distinction that has behavioral repercussions. Participants were shown displays of dots and asked to evaluate sentences with each, every, or all (...) combined with a predicate. We find that participants are better at estimating how many things the predicate applied to after evaluating sentences in which universal quantification is indicated with every or all, as opposed to each. We argue that every and all are understood in second-order terms that encourage group representation, while each is understood in first-order terms that encourage individual representation. Since the sentences that participants evaluate are truth-conditionally equivalent, our results also bear on questions concerning how meanings are related to truth-conditions. (shrink)

The category \(\mathbb {DRDL}{'}\), whose objects are c-differential residuated distributive lattices satisfying the condition \(\textbf{CK}\), is the image of the category \(\mathbb {RDL}\), whose objects are residuated distributive lattices, under the categorical equivalence \(\textbf{K}\) that is constructed in Castiglioni et al. (Stud Log 90:93–124, 2008). In this paper, we introduce weak monadic residuated lattices and study some of their subvarieties. In particular, we use the functor \(\textbf{K}\) to relate the category \(\mathbb {WMRDL}\), whose objects are weak monadic residuated distributive (...) lattices, and the category \(\mathbb {WMDRDL}{'}\), whose objects are pairs formed by an object of \(\mathbb {DRDL}{'}\) and a center weak universal quantifier. (shrink)

A computable graph is constructed which is not computably isomorphic to any polynomial-time graph with a standard universe . Conditions are given under which a computable graph is computably isomorphic to a polynomial-time graph with a standard universe — for example, if every vertex has finite degree. Two special types of graphs are studied. It is shown that any computable tree is recursively isomorphic to a p-time tree with standard universe and that any computable equivalence relation is computably (...) isomorphic to a p-time equivalence relation with standard universe. (shrink)