Let ω be the set of natural numbers. For functions f, g: ω → ω, we say f is dominated by g if f < g for all but finitely many n ∈ ω. We consider the standard “fair coin” probability measure on the space 2ω of in-finite sequences of 0's and 1's. A Turing oracle B is said to be almost everywhere dominating if, for measure 1 many X ∈ 2ω, each function which is Turing computable from X is (...) dominated by some function which is Turing computable from B. Dobrinen and Simpson have shown that the almost everywhere domination property and some of its variant properties are closely related to the reverse mathematics of measure theory. In this paper we exposit some recent results of Kjos-Hanssen, Kjos-Hanssen/Miller/Solomon, and others concerning LR-reducibility and almost everywhere domination. We also prove the following new result: If B is almost everywhere dominating, then B is superhigh, i. e., 0″ is truth-table computable from B ′, the Turing jump of B. (shrink)

To each computable enumerable (c.e.) set A with a particular enumeration {As}s∈ω, there is associated a settling function mA(x), where mA(x) is the last stage when a number less than or equal to x was enumerated into A. One c.e. set A is settling time dominated by another set B (B >st A) if for every computable function f, for all but finitely many x, mB(x) > f(m₄(x)). This settling-time ordering, which is a natural extension to an ordering of the (...) idea of domination, was first introduced by Nabutovsky and Weinberger in [3] and Soare [6]. They desired a sequence of sets descending in this relationship to give results in differential geometry. In this paper we examine properties of the <st ordering. We show that it is not invariant under computable isomorphism, that any countable partial ordering embeds into it, that there are maximal and minimal sets, and that two c.e. sets need not have an inf or sup in the ordering. We also examine a related ordering, the strong settling-time ordering where we require for all computable f and g, for almost all x, mB(x) > f(mA(g(x))). (shrink)

We study the relationship between effective domination properties and the bounded jump. We answer two open questions about the bounded jump: We prove that the analogue of Sacks jump inversion fails for the bounded jump and the wtt-reducibility. We prove that no c.e. bounded high set can be low by showing that they all have to be Turing complete. We characterize the class of c.e. bounded high sets as being those sets computing the Halting problem via a reduction (...) with use bounded by an ω-c.e. function. We define several notions of a c.e. set being effectively dominant, and show that together with the bounded high sets they form a proper hierarchy. (shrink)

Say that α is an n-strongly c. e. real if α is a sum of n many strongly c. e. reals, and that α is regular if α is n-strongly c. e. for some n. Let Sn be the set of all n-strongly c. e. reals, Reg be the set of regular reals and CE be the set of c. e. reals. Then we have: S1 ⊂ S2 ⊂ · · · ⊂ Sn ⊂ · · · ⊂ ⊂ Reg (...) ⊂ CE. This gives a hierarchy of the c. e. reals. We also study the regularity of the d. c. e. reals. (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 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)

Sometimes dictators are benevolent. Sometimes masters are kind and gentle to their slaves. John Adams was a pretty good "husband" to Abigail Adams. But it seems like there’s something very wrong with being a dictator or a master or a spouse with the power that John Adams had over Abigail Adams in late 18th Century America. A theory of domination tries to pinpoint what’s distinctive about dictatorship and mastery and traditional husbanding, and what is distinctively wrong with such—even the (...) benevolent, kind, gentle, and pretty good varieties. -/- There has been a lot of thinking about domination over the last twenty-five or so years. This is due largely to the efforts of republican political philosophers, who have used domination and its absence — nondomination — as the primary moving part in their conceptions of freedom. Even so, perhaps because domination has often appeared in a supporting role for the analysis of other concepts, conceptualizing domination itself is still a fairly nascent endeavor. -/- Accounts of domination need at least two movements. First, domination is a subset of a social phenomenon with—at least more or less—unobjectionable varieties. Maybe this social phenomenon is power; maybe it’s the capacity to interfere with choice; maybe something else. Whatever it is, it’s plausible to think it’s not always bad. Sometimes it’s perfectly alright to have power over someone else; sometimes there’s nothing wrong with the capacity to interfere with someone’s choices. (For example, if what you’d rather do is go after your neighbor with a meat-axe, it’s a good thing if someone is in a position to interfere with you, or if your choice situation can be worsened, so that charging next door, axe in hand, without having to worry about a lengthy prison sentence is not among your options.) The first task of a theory of domination is to specify exactly what this broader social phenomenon is. -/- Next we need to figure out what gives domination its negative moral valence, and thus sets it apart from that broader social phenomenon. Republicans usually say that what makes domination morally problematic is its arbitrariness. Frank Lovett claims that domination is instantiated in a social relationship between A and B only if A is permitted to exercise power over B arbitrarily (2010, 120). Philip Pettit, until recently, claimed that domination is a capacity for arbitrary interference. Marilyn Friedman calls it “arbitrary interference in someone’s choices…”(2008, 265). If that’s right, and arbitrariness is the difference maker between forms of social power that aren’t domination and those that are, we need a story about what arbitrariness is. -/- Such are the two essential tasks of conceptualizing domination. I offer here what I think are important course-corrections for recent attempts to complete them. In §1 I set out some desiderata for a useful conception of domination before sketching the elements of earlier attempts I think we can’t do without. §2 addresses the problem of “cheap domination”. This is the problem of identifying ordinary, innocuous human interactions with anything we call domination. I don’t think we should do this, but some of the most influential accounts of domination on offer nowadays — in particular, Lovett’s and Pettit’s — have this result. Instead, I contend that only certain forms of social power can underwrite domination, and try to provide a principled way of identifying what these forms are like. Finally, in §3, I address the nature of arbitrariness. (shrink)

Supervenience in most of its guises entails necessary coextension. Thus theoretical supervenience entails nomically necessary coextension. Kim's result, thus strengthened, has yet to hit home. I suspect that many supervenience enthusiasts would cool at necessary coextension: they didn't mean to be saying anything quite so strong. Furthermore, nomically necessary coextension can be a good reason for property identification, leading to reducibility in principle. This again is more than many supervenience theorists bargained for. They wanted supervenience without reducibility. It (...) is not always available for this mediating role. (shrink)

Let , be a sequence of pseudo-metric spaces, and let p≥1. For , let . For Borel reducibility between equivalence relations , we show it is closely related to finitely Hölder embeddability between pseudo-metric spaces.

Freedom as non-domination provides a distinctive criterion for assessing the justifiability of migration controls, different from both freedom of movement and autonomy. Migration controls are dominating insofar as they threaten to coerce potential migrants. Both the general right of states to control migration, and the wide range of discretionary procedures prevalent in migration controls, render outsiders vulnerable to arbitrary power. While the extent and intensity of domination varies, it is sufficient under contemporary conditions of globalization to warrant limits (...) on states’ discretion with respect to admission. Reducing domination requires, rather than removing all immigration restrictions or democratically justifying them to all, that there be certain constraints on states’ freedom to control migration: giving migrants a publicly secured status somewhat analogous to that enjoyed by citizens, subjecting migration controls to higher legal regulation, and making immigration policies and decision contestable by those who are subject to them. (shrink)

First, language and axioms of Church's paper 'Comparison of Russell's Resolution of the Semantical Antinomies with that of Tarski' are slightly modified and a version of the Liar paradox tentatively reconstructed. An obvious natural solution of the paradox leads to a hierarchy of truth predicates which is of a different kind from the one defined by Church: it depends on the enlargement of the semantical vocabulary and its levels do not differ in the ramified-type-theoretical sense. Second, two attempts are made (...) in order to justify the Russellian, and perhaps Churchian, idea that language should not be fragmented beyond what is required by type distinctions. After all, because of reducibility, which seems to allow a semantics without propositions, this comes out to be possible only at the cost of resorting to two disputable theses. (shrink)

Rosenberg’s general argumentative strategy in favour of panpsychism is an extension of a traditional pattern. Although his argument is complex and intricate, I think a model that is historically significant and fundamentally similar to the position Rosenberg advances might help us understand the case for panpsychism. Thus I want to begin by considering a Leibnizian argument for panpsychism.

We introduce a reducibility preordering between classes of countable structures, each class containing only structures of a given similarity type (which is allowed to vary from class to class). Though we sometimes work in a slightly larger context, we are principally concerned with the case where each class is an invariant Borel class (i.e. the class of all models, with underlying set $= \omega$, of an $L_{\omega_1\omega}$ sentence; from this point of view, the reducibility can be thought of (...) as a (rather weak) sort of $L_{\omega_1\omega}$-interpretability notion). We prove a number of general results about this notion, but our main thrust is to situate various mathematically natural classes with respect to the preordering, most notably classes of algebraic structures such as groups and fields. (shrink)

Theories of domination are primarily attempts to understand the value of justice, freedom, and equality by examining cases where they are absent. Such theories seek to clarify and systematize our judgments about what it is to be weak against uncontrolled strength, i.e., about what it is to be vulnerable, degraded, and defenseless against unrestrained power. -/- Much contemporary disagreement about domination involves competing answers to three questions: (1) Who, or what, can dominate? (2) Is it possible to dominate (...) merely by having power with a certain structure, or is domination an exercise or an abuse of power? (3) Exercised or unexercised, what kind of power is domination? The remainder of this entry will address each of these questions in turn, then conclude with a survey of how the idea of domination has been used in recent applied ethical theory. It will become clear as we examine competing answers to these three questions that different theorists have very different ideas of why, exactly, we need a theory of domination. There may be wide agreement that we need the idea of domination to make sense of unjust power relations, but unjust power relations are wildly varied, and theorists of domination disagree not only about which varieties most need to be understood, but about how theorizing domination helps us to understand them. (shrink)

In Part One Leiss traces the idea of the domination of nature from the Renaissance to the nineteenth century.

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)

The investigation of computational properties of discontinuous functions is an important concern in computable analysis. One method to deal with this subject is to consider effective variants of Borel measurable functions. We introduce such a notion of Borel computability for single-valued as well as for multi-valued functions by a direct effectivization of the classical definition. On Baire space the finite levels of the resulting hierarchy of functions can be characterized using a notion of reducibility for functions and corresponding complete (...) functions. We use this classification and an effective version of a Selection Theorem of Bhattacharya-Srivastava in order to prove a generalization of the Representation Theorem of Kreitz-Weihrauch for Borel measurable functions on computable metric spaces: such functions are Borel measurable on a certain finite level, if and only if they admit a realizer on Baire space of the same quality. This Representation Theorem enables us to introduce a realizer reducibility for functions on metric spaces and we can extend the completeness result to this reducibility. Besides being very useful by itself, this reducibility leads to a new and effective proof of the Banach-Hausdorff-Lebesgue Theorem which connects Borel measurable functions with the Baire functions. Hence, for certain metric spaces the class of Borel computable functions on a certain level is exactly the class of functions which can be expressed as a limit of a pointwise convergent and computable sequence of functions of the next lower level. (shrink)

This paper is an investigation into what could be a goodexplication of ``theory S is reducible to theory T''''. Ipresent an axiomatic approach to reducibility, which is developedmetamathematically and used to evaluate most of the definitionsof ``reducible'''' found in the relevant literature. Among these,relative interpretability turns out to be most convincing as ageneral reducibility concept, proof-theoreticalreducibility being its only serious competitor left. Thisrelation is analyzed in some detail, both from the point of viewof the reducibility axioms and (...) of modal logic. (shrink)

With her conception of epistemic injustice, Miranda Fricker has opened up new normative dimensions for epistemology; that is, the injustice of denying one?s status as a knower. While her analysis of the remedies for such injustices focuses on the epistemic virtues of agents, I argue for the normative superiority of adapting a broadly republican conception of epistemic injustice. This argument for a republican epistemology has three steps. First, I focus on methodological and explanatory issues of identifying epistemic injustice and argue, (...) against Fricker, that identity prejudice fails to provide a sufficient explanatory basis for the spread and maintenance of such systematic epistemic injustice. Second, this systemic basis can be found not so much in the psychological attitudes of individual knowers, but in the relations of domination among groups and individuals in a society. Third, if such a presence of domination plays a primary explanatory role in all forms of epistemic injustice, it is likely that those who suffer from epistemic injustice will also suffer other forms of injustice and loss of status via the exercise of other forms of power and exclusion. (shrink)

Abstract: The article aims to sharpen the neo-republican contribution to international political thought by challenging Pettit’s view that only representative states may raise a valid claim to non-domination in their external relations. The argument proceeds in two steps: First I show that, conceptually speaking, the domination of states, whether representative or not, implies dominating the collective people at least in its fundamental, constitutive power. Secondly, the domination of states – and thus of their peoples – cannot be (...) justified normatively in the name of promoting individual non-domination because such a compensatory rationale misconceives the notion of domination in terms of a discrete exercise of power instead of as an ongoing power relation. This speaks in favour of a more inclusive law of peoples than Pettit (just as his liberal counterpart Rawls) envisages: In order to accommodate the claim of collective peoples to non-domination it has to recognize every state as a member of the international order. (shrink)

This paper develops a domination-based practice-dependent approach to justice, according to which it is practices of systemic domination which can be said to ground demands from justice. The domination-based approach developed overcomes the two most important objections levelled to alternative practice-dependent approaches. First, it eschews conservative implications and hence is immune to the status quo objection. Second, it is immune to the redundancy objection, which doubts whether empirical facts and practices can really play an irreducible role in (...) grounding justice. In theorising dominating practices in terms of practices of social power, a domination-based approach makes justice dependent on factual information in three ways: First, the principle of non-domination is indeterminate and can only be spelled out by taking into view particular contexts of domination. Second, the principle of non-domination is conditional on the existence of practices of social power. Third, social power possesses a structural ontology – to know whether A has social power over B we need to turn to social rules distributing agents' higher order status of normative authority towards each other. This explains in what way practices of social power – and of domination – are both factual and normative practices and hence how such practices are non-redundant in grounding justice. (shrink)

This chapter explores the different dimensions of domination, including whether it has a structural approach, its relation to race and imperialism, and how non-domination can be institutionalized and achieved at a global level.

The structure of the l-degrees included in an m-degree with a maximal set together with the l-reducibility relation is characterized. For this a special sublattice of the lattice of recursively enumerable sets under the set-inclusion is used.

In Europe and other regions of the world public debate concerning how many immigrants should be admitted, which rights those admitted should have, and which conditions can be required for access to citizenship is intense and enduring, and these have increasingly become central electoral issues. On the one hand, the harsh treatment of migrants is often a matter of public criticism; on the other hand, states are concerned about problems of welfare, security and social unrest that they have come to (...) associate with large-scale migration. At its most fundamental level, this debate concerns the question of how best to balance particularist principles of democratic self-determination and state sovereignty and universalist principles of individual freedom and human rights. The articles in this special issue examine whether the concept of domination can cast a distinctive light on the normative issues arising in this tension between state sovereignty and universal principles with respect to migration and the position of non-citizens in contemporary liberal democratic states. These issues arise both externally insofar as foreigners are subject to migration controls, and internally insofar as migrants live in states of which they are not full members. The issue thus examines the potential of domination as a concept to bring to bear on issues of migration controls, differential residence statuses and access to membership. (shrink)

The recent revival of civic republicanism has been grounded on a conception of liberty as non-domination. While this avenue of thought holds considerable promise, such a conception of liberty can only be as sound as the underlying concept of domination, and although the term appears frequently in the pages of contemporary political theory, unlike other basic concepts, domination has received remarkably little in the way of serious conceptual analysis. Indeed, one might be tempted to conclude that (...) class='Hi'>domination is not a substantive concept at all, but rather mere empty rhetoric. This essay represents an experiment of sorts to see if the concept of domination can indeed withstand sustained philosophical examination; to the extent that it is successful, it can be viewed as a friendly addition or amendment to the work of contemporary republican political theorists. (shrink)

Introduction A book with this title, dealing as it does with the political machinery and political education appropriate to a democratic society, ...

Let ${\cal T}$ be any of the three canonical truth theories CT^− (compositional truth without extra induction), FS^− (Friedman–Sheard truth without extra induction), or KF^− (Kripke–Feferman truth without extra induction), where the base theory of ${\cal T}$ is PA. We establish the following theorem, which implies that ${\cal T}$ has no more than polynomial speed-up over PA. Theorem.${\cal T}$is feasibly reducible to PA, in the sense that there is a polynomial time computable function f such that for every ${\cal T}$-proof (...) π of an arithmetical sentence ϕ, f(π) is a PA-proof of ϕ. (shrink)

This study builds on the work of contemporary civic republicans, supplying a detailed analysis of the concept of domination absent in the familiar accounts of political freedom as non-domination.

We study the Borel reducibility of isomorphism relations in the generalized Baire space math formula. In the main result we show for inaccessible κ, that if T is a classifiable theory and math formula is stable with the orthogonal chain property, then the isomorphism of models of T is Borel reducible to the isomorphism of models of math formula.

ABSTRACTThis paper assesses to what extent the neo-Republican accounts of Quentin Skinner and Philip Pettit adequately capture the nature of political liberty at Rome by focusing on Cicero's analysis of the libera res publica. Cicero's analysis in De Republica suggests that the rule of law and a modest menu of individual citizens’ rights guard against citizens being controlled by a master's arbitrary will, thereby ensuring the status of non-domination that constitutes freedom according to the neo-Republican view. He also shows (...) the difficulty of anchoring an argument for citizens’ full political participation in the value of non-domination. While Cicero believed such full participation was essential for a libera res publica, he, like other elite Romans, argued for participation on the basis of liberty conceived as the space to contend for and enhance one's social status. The sufficiency of the rule of law and citizens’ rights for securing a status of non-domination taken together with their insufficiency for ensuring a libera res publica suggests that neo-Republican accounts of liberty do not fully capture the idea as articulated in Cicero's Republicanism. (shrink)

Recent work in natural language semantics leads to some new observations on generalized quantifiers. In § 1 we show that English quantifiers of type $ $ are booleanly generated by their generalized universal and generalized existential members. These two classes also constitute the sortally reducible members of this type. Section 2 presents our main result--the Generalized Prefix Theorem (GPT). This theorem characterizes the conditions under which formulas of the form Q1x 1⋯ Qnx nRx 1⋯ xn and q1x 1⋯ qnx nRx (...) 1⋯ xn are logically equivalent for arbitrary generalized quantifiers Qi, qi. GPT generalizes, perhaps in an unexpectedly strong form, the Linear Prefix Theorem (appropriately modified) of Keisler & Walkoe (1973). (shrink)

After an initial period of feminist theorizing concerned with understanding patriarchy as a structure of male domination, many thinkers turned away from theorizing domination as such and focused instead on women's subjectivity, identity, and agency. While this has fostered important insights into the formation of women's preferences, desires, and choices, this focus on subjectivity and subject formation has largely overshadowed deeper understandings of patriarchy as a structure of male domination while producing elisions between agency and freedom. In (...) this article, I move to show how domination as a structural concept can help us to reclaim the idea of ‘patriarchy’ as a source of women's systematic oppression while freedom as non-domination, derived from early republican conceptualizations of freedom, can help us to disambiguate freedom from agency by taking as central the relative positions of actors within social and political structures. Structural freedom as non-domination is thus useful for feminist thinkers in that it gives us critical purchase on the dynamics inherent in unequal social and political relationships and can be linked clearly to the institutions and ideologies that shape and justify interactions between more powerful and less powerful groups. Further, from this point of view intersecting structures of domination can be analysed rather than intersecting identity categories, allowing us to take intersectionality into account and avoiding the need to ground feminist action on a unitary ‘category woman’. Finally, this analysis points toward the radical democratic connexion between freedom and participation in the creation of the material and symbolic structures that frame our collective lives. (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 study a strong enumeration reducibility, called bounded enumeration reducibility and denoted by ≤be, which is a natural extension of s-reducibility ≤s. We show that ≤s, ≤be, and enumeration reducibility do not coincide on the ${\Pi^0_1}$ –sets, and the structure ${\boldsymbol{\mathcal{D}_{\rm be}}}$ of the be-degrees is not elementarily equivalent to the structure of the s-degrees. We show also that the first order theory of ${\boldsymbol{\mathcal{D}_{\rm be}}}$ is computably isomorphic to true second order arithmetic: this answers an (...) open question raised by Cooper (Z Math Logik Grundlag Math 33:537–560, 1987). (shrink)

We prove the Compact Domination Conjecture for groups definable in linear o-minimal structures. Namely, we show that every definably compact group G definable in a saturated linear o-minimal expansion of an ordered group is compactly dominated by (G/G 00, m, π), where m is the Haar measure on G/G 00 and π : G → G/G 00 is the canonical group homomorphism.

We study the κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document}-Borel-reducibility of isomorphism relations of complete first order theories in a countable language and show the consistency of the following: For all such theories T and T′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^{\prime }$$\end{document}, if T is classifiable and T′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^{\prime }$$\end{document} is not, then the isomorphism of models of T′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} (...) \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T^{\prime }$$\end{document} is strictly above the isomorphism of models of T with respect to κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document}-Borel-reducibility. In fact, we can also ensure that a range of equivalence relations modulo various non-stationary ideals are strictly between those isomorphism relations. The isomorphism relations are considered on models of some fixed uncountable cardinality obeying certain restrictions. (shrink)