The isomorphism invariance criterion of logical nature has much to commend it. It can be philosophically motivated by the thought that logic is distinctively general or topic neutral. It is capable of precise set-theoretic formulation. And it delivers an extension of ‘logical constant’ which respects the intuitively clear cases. Despite its attractions, the criterion has recently come under attack. Critics such as Feferman, MacFarlane and Bonnay argue that the criterion overgenerates by incorrectly judging mathematical notions as logical. We (...) consider five possible precisifications of the overgeneration argument and find them all unconvincing. (shrink)

This is a survey of work on set-theoretical invariance criteria for logicality. It begins with a review of the Tarski-Sher thesis in terms, first, of permutation invariance over a given domain and then of isomorphism invariance across domains, both characterized by McGee in terms of definability in the language L∞,∞. It continues with a review of critiques of the Tarski-Sher thesis, and a proposal in response to one of those critiques via homomorphism invariance. That has (...) quite divergent characterization results depending on its formulation, one in terms of FOL, the other by Bonnay in terms of L∞,∞, both without equality. From that we move on to a survey of Bonnay’s work on similarity relations between structures and his results that single out invariance with respect to potential isomorphism among all such. Turning to the critique that calls for sameness of meaning of a logical operation across domains, the paper continues with a result showing that the isomorphism invariant operations that are absolutely definable with respect to KPU−Inf are exactly those definable in full FOL; this makes use of an old theorem of Manders. The concluding section is devoted to a critical discussion of the arguments for set-theoretical criteria for logicality. (shrink)

An isomorphism is built between the separable complex Hilbert space (quantum mechanics) and Minkowski space (special relativity) by meditation of quantum information (i.e. qubit by qubit). That isomorphism can be interpreted physically as the invariance between a reference frame within a system and its unambiguous counterpart out of the system. The same idea can be applied to Poincaré’s conjecture (proved by G. Perelman) hinting another way for proving it, more concise and meaningful physically. Mathematically, the isomorphism (...) means the invariance to choice, the axiom of choice, well-ordering, and well-ordering “theorem” (or “principle”) and can be defined generally as “information invariance”. (shrink)

A serious crisis is identified in theories of neurocomputation, marked by a persistent disparity between the phenomenological or experiential account of visual perception and the neurophysiological level of description of the visual system. In particular, conventional concepts of neural processing offer no explanation for the holistic global aspects of perception identified by Gestalt theory. The problem is paradigmatic and can be traced to contemporary concepts of the functional role of the neural cell, known as the Neuron Doctrine. In the absence (...) of an alternative neurophysiologically plausible model, I propose a perceptual modeling approach, to model the percept as experienced subjectively, rather than modeling the objective neurophysiological state of the visual system that supposedly subserves that experience. A Gestalt Bubble model is presented to demonstrate how the elusive Gestalt principles of emergence, reification, and invariance can be expressed in a quantitative model of the subjective experience of visual consciousness. That model in turn reveals a unique computational strategy underlying visual processing, which is unlike any algorithm devised by man, and certainly unlike the atomistic feed-forward model of neurocomputation offered by the Neuron Doctrine paradigm. The perceptual modeling approach reveals the primary function of perception as that of generating a fully spatial virtual-reality replica of the external world in an internal representation. The common objections to this picture-in-the-head concept of perceptual representation are shown to be ill founded. Key Words: brain-anchored; Cartesian theatre; consciousness; emergence; extrinsic constraints; filling-in; Gestalt; homunculus; indirect realism; intrinsic constraints; invariance; isomorphism; multistability; objective phenomenology; perceptual modeling; perspective; phenomenology; psychophysical parallelism; psychophysical postulate; qualia; reification; representationalism; structural coherence. (shrink)

The suggestion that analytic isomorphism should be rejected applies especially to the domain of speech perception because (1) the guiding assumption that solving the lack of invariance problem is the key to explaining speech perception is a form of analytic isomorphism, and (2) after nearly half a century of research there is virtually no empirical evidence of isomorphism between perceptual experience and lower-level processing units.

According to the classical invariance criterion, a term is logical if and only if its extension is isomorphism-invariant. However, a number of authors have devised examples that challenge the sufficiency of this condition: accepting these examples as logical constants would introduce objectionable contingent elements into logic. Recently, Gil Sagi has responded that these objections are based on a fallacious inference from the modal status of a sentence to the modal status of the proposition expressed by that sentence. The (...) present paper demonstrates that Sagi’s response, though successful, is futile. There is another objection, based on the same type of example, that is not susceptible to Sagi’s criticism: accepting the examples as logical terms would have the fatal consequence that any contingent metalanguage sentence is entailed by the truth of some logically true object-language sentence. I conclude with a sketch of an alternative to the classical invariance criterion. (shrink)

We will present several results on two types of continuous models of -calculus, namely graph models and extensional models. By introducing a variant of Engeler's model construction, we are able to generalize the results of [7] and to give invariants that determine a large family of graph models up to applicative isomorphism. This covers all graph models considered in the litterature so far. We indicate briefly how these invariants may be modified in order to determine extensional models as well.Furthermore, (...) we use our construction to exhibit graph models that are not equationally equivalent. We indicate once again how the construction passes on to extensional models. (shrink)

What is a logical constant? The question is addressed in the tradition of Tarski's definition of logical operations as operations which are invariant under permutation. The paper introduces a general setting in which invariance criteria for logical operations can be compared and argues for invariance under potential isomorphism as the most natural characterization of logical operations.

The essay discusses a recurrent criticism of the isomorphism-invariance criterion for logical terms, according to which the criterion pertains only to the extension of logical terms, and neglects the meaning, or the way the extension is fixed. A term, so claim the critics, can be invariant under isomorphisms and yet involve a contingent or a posteriori component in its meaning, thus compromising the necessity or apriority of logical truth and logical consequence. This essay shows that the arguments underlying (...) the criticism are flawed since they rely on an invalid inference from the modal or epistemic status of statements in the metalanguage to that of statements in the object-language. The essay focuses on McCarthy’s version of the argument, but refers to Hanson and McGee’s versions as well. (shrink)

Tarski defined a way of assigning to each Boolean algebra, B, an invariant inv(B) ∈ In, where In is a set of triples from ℕ, such that two Boolean algebras have the same invariant if and only if they are elementarily equivalent. Moreover, given the invariant of a Boolean algebra, there is a computable procedure that decides its elementary theory. If we restrict our attention to dense Boolean algebras, these invariants determine the algebra up to isomorphism. In this paper (...) we analyze the complexity of the question "Does B have invariant x?" For each x ∈ In we define a complexity class Γx that could be either Σⁿ, Πⁿ, Σⁿ ∧ Πⁿ, or Πω+1 depending on x, and we prove that the set of indices for computable Boolean algebras with invariant x is complete for the class Γx. Analogs of many of these results for computably enumerable Boolean algebras were proven in earlier works by Selivanov. In a more recent work, he showed that similar methods can be used to obtain the results for computable ones. Our methods are quite different and give new results as well. As the algebras we construct to witness hardness are all dense, we establish new similar results for the complexity of various isomorphism problems for dense Boolean algebras. (shrink)

The paper examines the invariance principles proposed by the analytical and phenomenological traditions for demarcating the boundaries of formal and regional ontologies. The principle of invariance with respect to isomorphic transformations, generalizing Alfred Tarski’s criterion for logical concepts, is extended to formal ontology as the theory of manifolds in its phenomenological interpretation. Isomorphism types, which are abstract individuals of the highest order, hypostases of forms of all possible ontologies, are considered as model-theoretical analogs of manifolds. The correlativity (...) of the phenomenological principles of demarcation of ontological regions and the criterion of invariance with respect to isomorphic transformations is demonstrated, leading to the convergence of logic and formal mathematics in both traditions and, at the same time, to the exclusion of geometry from formal ontology. Particular attention is paid to the discussion of the analytical and phenomenological traditions of the synthetic (material) a priori and the contribution that Wittgenstein’s doctrine on internal relations at different stages of its evolution makes to this discussion. The paper reveals the basis for later Wittgenstein’s criticism of his earlier project of creating a phenomenological language for expressing internal regional relations (such as the colours exclusion). The paper shows how Wittgenstein’s doubts about the possibility of an ideal notation based on the dichotomy of the logical and phenomenological led him to the study of invariants arising in language games. These invariants are not determined by the special properties of the categorical objects of formal ontology or the structures of regional subject matter, but are stable equilibria generated by the “consensus of actions.” New perspectives are outlined that are opened for logic and phenomenology by switching focus from the invariants of ontological structures to the invariants of structured interactions of various agents. (shrink)

In this paper, I argue that the Hole Argument can be formulated without using the notion of isomorphism, and for this reason it is not threatened by the criticism of Halvorson and Manchak (Br J Philos Sci, 2022. https://doi.org/10.1086/719193). Following Earman and Norton (Br J Philos Sci 38, pp. 515–525, 1987), I divide the Hole Argument into two steps: the proof of the Gauge Theorem and the transition from the Gauge Theorem to the conclusion of radical indeterminism. In the (...) analaysis of the first step, I argue that the Gauge Theorem does not rely on the notion of isomorphism but on the notion of the diffeomorphism-invariance of the equations of local spacetime theories; however, for this approach to work, the definition of local spacetime theories needs certain amendments with respect to Earman and Norton’s formulation. In the analysis of the second step, I postulate that we should use the notion of radical indeterminism instead of indeterminism simpliciter and that we should not decide in advance what kind of maps are to be used in comparing models. Instead, we can tentatively choose some kind of maps for this purpose and check whether a given choice leads to radical indeterminism involving empirically indistinguishable models. In this way, the use of the notion of isomorphism is also avoided in the second step of the Hole Argument. A general picture is that physical equivalence can be established by means of an iterative procedure in which we examine various candidate classes of maps, and, depending on the outcomes, we need to broaden or narrow these classes. The Hole Argument can be viewed as a particular instance of this procedure. (shrink)

The language of standard propositional modal logic has one operator (? or ?), that can be thought of as being determined by the quantifiers ? or ?, respectively: for example, a formula of the form ?F is true at a point s just in case all the immediate successors of s verify F.This paper uses a propositional modal language with one operator determined by a generalized quantifier to discuss a simple connection between standard invariance conditions on modal formulas and (...) generalized quantifiers: the combined generalized quantifier conditions of conservativity and extension correspond to the modal condition of invariance under generated submodels, and the modal condition of invariance under bisimulations corresponds to the generalized quantifier being a Boolean combination of ? and ? (shrink)

We consider several variants of Keisler’s isomorphism theorem. We separate these variants by showing implications between them and cardinal invariants hypotheses. We characterize saturation hypotheses that are stronger than Keisler’s theorem with respect to models of size $\aleph _1$ and $\aleph _0$ by $\mathrm {CH}$ and $\operatorname {cov}(\mathsf {meager}) = \mathfrak {c} \land 2^{<\mathfrak {c}} = \mathfrak {c}$ respectively. We prove that Keisler’s theorem for models of size $\aleph _1$ and $\aleph _0$ implies $\mathfrak {b} = \aleph _1$ and (...) $\operatorname {cov}(\mathsf {null}) \le \mathfrak {d}$ respectively. As a consequence, Keisler’s theorem for models of size $\aleph _0$ fails in the random model. We also show that for Keisler’s theorem for models of size $\aleph _1$ to hold it is not necessary that $\operatorname {cov}(\mathsf {meager})$ equals $\mathfrak {c}$. (shrink)

Let [Formula: see text] be a monster model of an arbitrary theory [Formula: see text], let [Formula: see text] be any tuple of bounded length of elements of [Formula: see text], and let [Formula: see text] be an enumeration of all elements of [Formula: see text]. By [Formula: see text] we denote the compact space of all complete types over [Formula: see text] extending [Formula: see text], and [Formula: see text] is defined analogously. Then [Formula: see text] and [Formula: see (...) text] are naturally [Formula: see text]-flows. We show that the Ellis groups of both these flows are of bounded size, providing an explicit bound on this size. Next, we prove that these Ellis groups do not depend on the choice of the monster model [Formula: see text]; thus, we say that these Ellis groups are absolute. We also study minimal left ideals of the Ellis semigroups of the flows [Formula: see text] and [Formula: see text]. We give an example of a NIP theory in which the minimal left ideals are of unbounded size. Then we show that in each of these two cases, boundedness of a minimal left ideal is an absolute property and that whenever such an ideal is bounded, then in some sense its isomorphism type is also absolute. Under the assumption that [Formula: see text] has NIP, we give characterizations of when a minimal left ideal of the Ellis semigroup of [Formula: see text] is bounded. Then we adapt the proof of Theorem 5.7 in Definably amenable NIP groups, J. Amer. Math. Soc. 31 609–641 to show that whenever such an ideal is bounded, a certain natural epimorphism 863–932]) from the Ellis group of the flow [Formula: see text] to the Kim–Pillay Galois group [Formula: see text] is an isomorphism. We also obtain some variants of these results, formulate some questions, and explain differences which occur when the flow [Formula: see text] is replaced by [Formula: see text]. (shrink)

It follows directly from Shelah’s structure theory that if T is a classifiable theory, then the isomorphism type of any model of T is determined by the theory of that model in the language L∞,ω1. Leo Harrington asked if one could improve this to the logic L∞, In [S. Shelah, Characterizing an -saturated model of superstable NDOP theories by its L∞,-theory, Israel Journal of Mathematics 140 61–111] Shelah gives a partial positive answer, showing that for T a countable superstable (...) NDOP theory, two -saturated models of T are isomorphic if and only if they have the same L∞,-theory. We give here a negative answer to the general question by constructing two classifiable theories, each with 21 pairwise non-isomorphic models of cardinality 1, which are all L∞,-equivalent, a shallow depth 3 ω-stable theory and a shallow NOTOP depth 1 superstable theory. In the other direction, we show that in the case of an ω-stable depth 2 theory, the L∞,-theory is enough to describe the isomorphism type of all models. (shrink)

To understand logic is, first and foremost, to understand logical consequence. This Element provides an in-depth, accessible, up-to-date account of and philosophical insight into the semantic, model-theoretic conception of logical consequence, its Tarskian roots, and its ideas, grounding, and challenges. The topics discussed include: the passage from Tarski's definition of truth to his definition of logical consequence, the need for a non-proof-theoretic definition, the idea of a semantic definition, the adequacy conditions of preservation of truth, formality, and necessity, the nature, (...) structure, and totality of models, the logicality problem that threatens the definition of logical consequence, a general solution to the logicality, formality, and necessity problems/challenges, based on the isomorphism-invariance criterion of logicality, philosophical background and justification of the isomorphism-invariance criterion, and major criticisms of the semantic definition and the isomorphism-invariance criterion. (shrink)

Aimed at graduate students, research logicians and mathematicians, this much-awaited text covers over 40 years of work on relative classification theory for nonstandard models of arithmetic. The book covers basic isomorphism invariants: families of type realized in a model, lattices of elementary substructures and automorphism groups.

The standard view about generic generalizations is that they have a tripartite quantificational logical form involving a phonologically null quantificational expression called ‘Gen’. However, proponents of the cognitive defaults theory of generics have forcefully rejected this view, instead arguing that generics express the default generalizations of our cognitive system, and, as such, they are different in kind from quantificational generalizations. While extant criticism of the cognitive defaults theory has focused on the extent to which it is supported by the empirical (...) evidence, there has been little discussion of a neglected, albeit essential, theoretical argument in its defence, namely, that generics cannot be quantificational because they lack a central logical property of quantifiers: isomorphism invariance. This paper addresses this lacuna by considering and rejecting this argument. Consequently, an essential argument in favour of the cognitive default theory is found wanting. (shrink)

This note argues against Barwise and Etchemendy's claim that their semantics for self-reference requires use of Aczel's anti-foundational set theory, AFA, semantics for self-reference requires use of Aczel's anti-foundational set theory, AFA, ones irrelevant to the task at hand" (The Liar, p. 35). Switching from ZF to AFA neither adds nor precludes any isomorphism types of sets. So it makes no difference to ordinary mathematics. I argue against the author's claim that a certain kind of 'naturalness' nevertheless makes AFA (...) preferable to ZF for their purposes. I cast their semantics in a natural, isomorphism invariant form with self-reference as a fixed point property for propositional operators. Independent of the particulars of any set theory, this form is somewhat simpler than theirs and easier to adapt to other theories of self-reference. (shrink)

I attempt an explication of what it means for an operation across domains to be the same on all domains, an issue that ) took to be central for a successful delimitation of the logical operations. Some properties that seem strongly related to sameness are examined, notably isomorphism invariance, and sameness under extensions of the domain. The conclusion is that although no precise criterion can satisfy all intuitions about sameness, combining the two properties just mentioned yields a reasonably (...) robust and useful explication of sameness across domains. (shrink)

We prove a number of results concerning the variety of first-order theories and isomorphism types of pairs of the form $(N,M)$ , where $N$ is a countable recursively saturated model of Peano Arithmetic and $M$ is its cofinal submodel. We identify two new isomorphism invariants for such pairs. In the strongest result we obtain continuum many theories of such pairs with the fixed greatest common initial segment of $N$ and $M$ and fixed lattice of interstructures $K$ , such (...) that $M\prec K\prec N$. (shrink)

A P-point ultrafilter over $\omega$ is called an interval P-point if for every function from $\omega$ to $\omega$ there exists a set _A_ in this ultrafilter such that the restriction of the function to _A_ is either a constant function or an interval-to-one function. In this paper we prove the following results. (1) Interval P-points are not isomorphism invariant under $\textsf{CH}$ or $\textsf{MA}$. (2) We identify a cardinal invariant $\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})$ such that every filter (...) base of size less than continuum can be extended to an interval P-point if and only if $\textbf{non}^{**}({\mathcal {I}}_{\tiny {\hbox {int}}})={\mathfrak {c}}$. (3) We prove the generic existence of slow/rapid non-interval P-points and slow/rapid interval P-points which are neither quasi-selective nor weakly Ramsey under the assumption ${\mathfrak {d}}={\mathfrak {c}}$ or $\textbf{cov}({\mathcal {B}})={\mathfrak {c}}$. (shrink)

We show, roughly speaking, that it requires ω iterations of the Turing jump to decode nontrivial information from Boolean algebras in an isomorphism invariant fashion. More precisely, if α is a recursive ordinal, A is a countable structure with finite signature, and d is a degree, we say that A has αth-jump degree d if d is the least degree which is the αth jump of some degree c such there is an isomorphic copy of A with universe ω (...) in which the functions and relations have degree at most c. We show that every degree d ≥ 0 (ω) is the ωth jump degree of a Boolean algebra, but that for $n no Boolean algebra has nth-jump degree $\mathbf{d} > 0^{(n)}$ . The former result follows easily from work of L. Feiner. The proof of the latter result uses the forcing methods of J. Knight together with an analysis of various equivalences between Boolean algebras based on a study of their Stone spaces. A byproduct of the proof is a method for constructing Stone spaces with various prescribed properties. (shrink)

A relational structure \ is called reversible iff each bijective homomorphism from \ onto \ is an isomorphism, and linear orders are prototypical examples of such structures. One way to detect new reversible structures of a given relational language L is to notice that the maximal or minimal elements of isomorphism-invariant sets of interpretations of the language L on a fixed domain X determine reversible structures. We isolate certain syntactical conditions providing that a satisfiable \-theory defines a class (...) of interpretations having extreme elements on a fixed domain and detect several classes of reversible structures. For some of these classes, we characterize the corresponding reversible extreme interpretations. In particular, we characterize the reversible countable ultrahomogeneous graphs. (shrink)

This paper deals with the adequacy of the model-theoretic definition of logical consequence. Logical consequence is commonly described as a necessary relation that can be determined by the form of the sentences involved. In this paper, necessity is assumed to be a metaphysical notion, and formality is viewed as a means to avoid dealing with complex metaphysical questions in logical investigations. Logical terms are an essential part of the form of sentences and thus have a crucial role in determining logical (...) consequence. Gila Sher and Stewart Shapiro each propose a formal criterion for logical terms within a model-theoretic framework, based on the idea of invariance under isomorphism. The two criteria are formally equivalent, and thus we have a common ground for evaluating and comparing Sher and Shapiro philosophical justification of their criteria. It is argued that Shapiro's blended approach, by which models represent possible worlds under interpretations of the language, is preferable to Sher’s formal-structural view, according to which models represent formal structures. The advantages and disadvantages of both views’ reliance on isomorphism are discussed. (shrink)

Logics L F (M) are considered, in which M ("most") is a new first-order quantifier whose interpretation depends on a given filter F of subsets of ω. It is proved that countable compactness and axiomatizability are each equivalent to the assertion that F is not of the form $\{(\bigcap F) \cup X:|\omega - X| with $|\omega - \bigcap F| = \omega$ . Moreover the set of validities of L F (M) and even of L F ω 1 ω (M) depends (...) only on a few basic properties of F. Similar characterizations are given of the class of filters F for which L F (M) has the interpolation or Robinson properties. An omitting types theorem is also proved. These results sharpen the corresponding known theorems on weak models (U, q), where the collection q is allowed to vary. In addition they provide extensions of first-order logic which possess some nice properties, thus escaping from contradicting Lindstrom's Theorem [1969] only because satisfaction is not isomorphism-invariant (as it is tied to the filter F). However, Lindstrom's argument is applied to characterize the invariant sentences as just those of first-order logic. (shrink)

A brief article introducing *One True Logic*. The book argues that there is one correct foundational logic and that it is highly infinitary.

Ulm’s Theorem presents invariants that classify countable abelian torsion groups up to isomorphism. Barwise and Eklof extended this result to the classification of arbitrary abelian torsion groups up to L∞ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{\infty \omega }$$\end{document}-equivalence. In this paper, we extend this classification to a class of mixed Zp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {Z}}_p$$\end{document}-modules which includes all Warfield modules and is closed under L∞ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} (...) \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{\infty \omega }$$\end{document}-equivalence. The defining property of these modules is the existence of what we call a partial decomposition basis, a generalization of the concept of decomposition basis. We prove a complete classification theorem in L∞ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{\infty \omega }$$\end{document} using invariants deduced from the classical Ulm and Warfield invariants. (shrink)

We discuss some aspects of the relation between dualities and gauge symmetries. Both of these ideas are of course multi-faceted, and we confine ourselves to making two points. Both points are about dualities in string theory, and both have the ‘flavour’ that two dual theories are ‘closer in content’ than you might think. For both points, we adopt a simple conception of a duality as an ‘isomorphism’ between theories: more precisely, as appropriate bijections between the two theories’ sets of (...) states and sets of quantities. The first point is that this conception of duality meshes with two dual theories being ‘gauge related’ in the general philosophical sense of being physically equivalent. For a string duality, such as T-duality and gauge/gravity duality, this means taking such features as the radius of a compact dimension, and the dimensionality of spacetime, to be ‘gauge’. The second point is much more specific. We give a result about gauge/gravity duality that shows its relation to gauge symmetries to be subtler than you might expect. For gauge theories, you might expect that the duality bijections relate only gauge-invariant quantities and states, in the sense that gauge symmetries in one theory will be unrelated to any symmetries in the other theory. This may be so in general; and indeed, it is suggested by discussions of Polchinski and Horowitz. But we show that in gauge/gravity duality, each of a certain class of gauge symmetries in the gravity/bulk theory, viz. diffeomorphisms, is related by the duality to a position-dependent symmetry of the gauge/boundary theory. (shrink)

The study contributes to building an understanding of the impact of political forces on the information environment of listed firms in a developing economy. Specifically, it investigates the tensions between politico-institutional factors and accounting regulation on the prolonged and incomplete implementation of the International Financial Reporting Standards in Bangladesh from 1998 to 2010. Two phases of interviews were conducted in 2010–2011 and IFRS-related enforcement documents from 1998 to 2010 were evaluated. The study contributes that IFRSs are being diffused to developing (...) countries like Bangladesh, but they invariably interact with local institutions, with variable outcomes. Coercive, normative and mimetic isomorphisms are low in Bangladesh. Notably, political forces have been undermining mimetic isomorphism because of the high level of government intervention and the high level of political lobbying. Political institutional pressures stand in the way of mimetic isomorphism and constitute negative forces that add further tension to accounting regulation in Bangladesh. Regarding the low level of normative isomorphism, there is evidence of a ‘blame culture’, with state institutions and professional accountancy institutions in the country blaming each other for the poor progress in IFRS implementation. Although the study focuses on Bangladesh, its results have implications for international policy makers, as well as the governments and regulators of other developing economies facing similar challenges in implementing IFRS. (shrink)

Using the theory of Borel equivalence relations we analyze the isomorphism relation on the countable models of a theory and develop a framework for measuring the complexity of possible complete invariants for isomorphism.

We introduce the notions of stationarily ordered types and theories; the latter generalizes weak o-minimality and the former is a relaxed version of weak o-minimality localized at the locus of a single type. We show that forking, as a binary relation on elements realizing stationarily ordered types, is an equivalence relation and that each stationarily ordered type in a model determines some order-type as an invariant of the model. We study weak and forking non-orthogonality of stationarily ordered types, show that (...) they are equivalence relations and prove that invariants of non-orthogonal types are closely related. The techniques developed are applied to prove that in the case of a binary, stationarily ordered theory with fewer than countable models, the isomorphism type of a countable model is determined by a certain sequence of invariants of the model. In particular, we confirm Vaught's conjecture for binary, stationarily ordered theories. (shrink)

We consider Borel equivalence relations E induced by actions of the infinite symmetric group, or equivalently the isomorphism relation on classes of countable models of bounded Scott rank. We relate the descriptive complexity of the equivalence relation to the nature of its complete invariants. A typical theorem is that E is potentially Π03 iff the invariants are countable sets of reals, it is potentially Π04 iff the invariants are countable sets of countable sets of reals, and so on. The (...) proofs use various techniques, including Vaught transforms, changing topologies, and the Scott analysis of countable models. (shrink)

The Univalence axiom, due to Vladimir Voevodsky, is often taken to be one of the most important discoveries arising from the Homotopy Type Theory research programme. It is said by Steve Awodey that Univalence embodies mathematical structuralism, and that Univalence may be regarded as ‘expanding the notion of identity to that of equivalence’. This article explores the conceptual, foundational and philosophical status of Univalence in Homotopy Type Theory. It extends our Types-as-Concepts interpretation of HoTT to Universes, and offers an account (...) of the Univalence axiom in such terms. We consider Awodey’s informal argument that Univalence is motivated by the principle that reasoning should be invariant under isomorphism, and we examine whether an autonomous and rigorous justification along these lines can be given. We consider two problems facing such a justification. First, there is a difference between equivalence and isomorphism and Univalence must be formulated in terms of the former. Second, the argument as presented cannot establish Univalence itself but only a weaker version of it, and must be supplemented by an additional principle. The article argues that the prospects for an autonomous justification of Univalence are promising. (shrink)

At the turn of the 21st century, topology, the mathematical study of spatial properties that remain the same under the continuous deformation of objects, has come to invest all fields of aesthetics and culture. In particular, the algebraic topology of continuity has added to the digital realm of binary information, the on and off states of 0s and 1s, an invariant property (e.g. a continuous function), which now governs the relation between different forms of data. As this invariant function of (...) continual transformation has entered the field of automated computation, the culture of binary digits has shifted towards a new level of calculation derived from the introduction of temporal quantities into finite sets of algorithmic instructions and parameters. This new level of topological computation, it will be argued, defines new operative procedures of control, constantly adding axioms at the limit of calculation through an invariant function that establishes a smooth or uninterrupted connectivity between distinct data. The establishment of a continual function between distinct forms of data is based on homeomorphism or topological isomorphism between data objects, of which parametricism, as the new global style for architecture and design, is a perfect example. (shrink)

The way, in which quantum information can unify quantum mechanics (and therefore the standard model) and general relativity, is investigated. Quantum information is defined as the generalization of the concept of information as to the choice among infinite sets of alternatives. Relevantly, the axiom of choice is necessary in general. The unit of quantum information, a qubit is interpreted as a relevant elementary choice among an infinite set of alternatives generalizing that of a bit. The invariance to the axiom (...) of choice shared by quantum mechanics is introduced: It constitutes quantum information as the relation of any state unorderable in principle (e.g. any coherent quantum state before measurement) and the same state already well-ordered (e.g. the well-ordered statistical ensemble of the measurement of the quantum system at issue). This allows of equating the classical and quantum time correspondingly as the well-ordering of any physical quantity or quantities and their coherent superposition. That equating is interpretable as the isomorphism of Minkowski space and Hilbert space. Quantum information is the structure interpretable in both ways and thus underlying their unification. Its deformation is representable correspondingly as gravitation in the deformed pseudo-Riemannian space of general relativity and the entanglement of two or more quantum systems. The standard model studies a single quantum system and thus privileges a single reference frame turning out to be inertial for the generalized symmetry [U(1)]X[SU(2)]X[SU(3)] “gauging” the standard model. As the standard model refers to a single quantum system, it is necessarily linear and thus the corresponding privileged reference frame is necessary inertial. The Higgs mechanism U(1) → [U(1)]X[SU(2)] confirmed enough already experimentally describes exactly the choice of the initial position of a privileged reference frame as the corresponding breaking of the symmetry. The standard model defines ‘mass at rest’ linearly and absolutely, but general relativity non-linearly and relatively. The “Big Bang” hypothesis is additional interpreting that position as that of the “Big Bang”. It serves also in order to reconcile the linear standard model in the singularity of the “Big Bang” with the observed nonlinearity of the further expansion of the universe described very well by general relativity. Quantum information links the standard model and general relativity in another way by mediation of entanglement. The linearity and absoluteness of the former and the nonlinearity and relativeness of the latter can be considered as the relation of a whole and the same whole divided into parts entangled in general. (shrink)

Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence, in terms of relatively simple invariants. Where this is impossible, it is useful to have concrete results saying so. In model theory and descriptive set theory, there is a large body of work showing that certain classes of mathematical structures admit classification while others do not. In the present paper, (...) we describe some recent work on classification in computable structure theory.Section 1 gives some background from model theory and descriptive set theory. From model theory, we give sample structure and non-structure theorems for classes that include structures of arbitrary cardinality. We also describe the notion of Scott rank, which is useful in the more restricted setting of countable structures. From descriptive set theory, we describe the basic Polish space of structures for a fixed countable language with fixed countable universe. We give sample structure and non-structure theorems based on the complexity of the isomorphism relation, and on Borel embeddings.Section 2 gives some background on computable structures. We describe three approaches to classification for these structures. The approaches are all equivalent. However, one approach, which involves calculating the complexity of the isomorphism relation, has turned out to be more productive than the others. Section 3 describes results on the isomorphism relation for a number of mathematically interesting classes—various kinds of groups and fields. In Section 4, we consider a setting similar to that in descriptive set theory. We describe an effective analogue of Borel embedding which allows us to make distinctions even among classes of finite structures. Section 5 gives results on computable structures of high Scott rank. Some of these results make use of computable embeddings. Finally, in Section 6, we mention some open problems and possible directions for future work. (shrink)

As part of his program to unify linear algebra and geometry using the language of Clifford algebra, David Hestenes has constructed a (well-known) isomorphism between the conformal group and the orthogonal group of a space two dimensions higher, thus obtaining homogeneous coordinates for conformal geometry.(1) In this paper we show that this construction is the Clifford algebra analogue of a hyperbolic model of Euclidean geometry that has actually been known since Bolyai, Lobachevsky, and Gauss, and we explore its wider (...) invariant theoretic implications. In particular, we show that the Euclidean distance function has a very simple representation in this model, as demonstrated by J. J. Seidel.(18). (shrink)

It is shown that by realizing the isomorphism features of the frequency and geometric interpretations of probability, Reichenbach comes very close to the idea of identifying mathematical probability theory with measure theory in his 1949 work on foundations of probability. Some general features of Reichenbach’s axiomatization of probability theory are pointed out as likely obstacles that prevented him making this conceptual move. The role of isomorphisms of Kolmogorovian probability measure spaces is specified in what we call the “Maxim of (...) Probabilism”, which states that a necessary condition for a concept to be probabilistic is its invariance with respect to measure-theoretic isomorphisms. The functioning of the Maxim of Probabilism is illustrated by the example of conditioning via conditional expectations. (shrink)

In this thesis, we study the complexity of some mathematical problems: in particular, those arising in computable analysis and algorithmic learning theory for algebraic structures. Our study is not limited to these two areas: indeed, in both cases, the results we obtain are tightly connected to ideas and tools coming from different areas of mathematical logic, including for example descriptive set theory and reverse mathematics.After giving the necessary preliminaries, we first study the uniform computational strength of the Cantor–Bendixson theorem in (...) the Weihrauch lattice. This work falls into the program connecting reverse mathematics and computable analysis via the framework of Weihrauch reducibility. We concentrate on problems related to perfect subsets of Polish spaces, studying the perfect set theorem, the Cantor–Bendixson theorem, and various problems arising from them. As far as we know, this is the first systematic study of problems at the level of $\mathbf {\Pi }^1_1\mathsf {-CA}_0$ in the Weihrauch lattice. We show that the strength of some of the problems we study depends on the topological properties of the Polish space under consideration, while others have the same strength once the space is rich enough.We continue considering problems related to (induced) subgraphs. We provide results on the (effective) Wadge complexity of sets of graphs, that are also used to determine the Weihrauch degree of certain decision problems. The decision problems we consider are defined for a fixed graph G, and they take as input a graph H, answering whether G is an (induced) subgraph of H: we also consider the opposite problem (i.e., answering whether H is an induced subgraph of G). We conclude this part on (induced) subgraphs considering the Weihrauch degree of “search problems.”These problems are defined for a fixed graph G, and they take as input a graph H such that G is an (induced) subgraph H: the output is a copy of G in H. In both cases, we highlight differences and analogies between the subgraph and the induced subgraph relation.We then move our attention to algorithmic learning theory, and we present the framework we use to study the learnability of families of algebraic structures: here, given a countable family of pairwise nonisomorphic structures $\mathfrak {K}$, a learner receives larger and larger pieces of an arbitrary copy of a structure in $\mathfrak {K}$ and, at each stage, is required to output a conjecture about the isomorphism type of such a structure. We say that $\mathfrak {K}$ is learnable if there exists a learner which eventually stabilizes to a correct guess. The framework was lacking a method for comparing the complexity of nonlearnable families, and so we propose a solution to this problem using tools coming from invariant descriptive set theory. To do so, we first prove that a family of structures is learnable if and only if its learning domain is continuously reducible to the relation $E_0$ of eventual agreement on infinite binary sequences and then, replacing $E_0$ with Borel equivalence relations of higher complexity, we obtain a new hierarchy of learning problems. This leads to the notion of E-learnability, where a family of structures $\mathfrak {K}$ is E-learnable, for a Borel equivalence relation E, if there is a continuous reduction from the isomorphism relation associated with $\mathfrak {K}$ to E. It is then natural to ask how the notion of E-learnability interacts with “classical” learning paradigms.Finally, we study the number of mind changes that a learner needs to learn a given family, both from a topological and a combinatorial point of view, and we consider how the complexity of a learner (in terms of Turing reducibility) affects the number of mind changes for learning a given family.Abstract prepared by Vittorio Cipriani.E-mail: vittorio.cipriani17@gmail.com.URL: https://air.uniud.it/handle/11390/1252226. (shrink)

The Wigner–Weyl mapping of quantum operators to classical phase space functions preserves the algebra, when operator multiplication is mapped to the binary “*” operation. However, this isomorphism is destroyed under the quasiclassical substitution of * with conventional multiplication; consequently, an approximate mapping is required if algebraic relations are to be preserved. Such a mapping is uniquely determined by the fundamental relations of quantum mechanics, as is shown in this paper. The resultant quasiclassical approximation leads to an algebraic derivation of (...) Thomas–Fermi theory, and a new quantization rule which—unlike semiclassical quantization—is non-invariant under action transformations of the Hamiltonian, in the same qualitative manner as the true eigenvalues. The quasiclassical eigenvalues are shown to be significantly more accurate than the corresponding semiclassical values, for a variety of 1D and 2D systems. In addition, certain standard refinements of semiclassical theory are shown to be easily incorporated into the quasiclassical formalism. (shrink)

We develop the theory of symmetry for a two-level quantum system in oder to illustrate the main ideas of the general theory of symmetry in quantum theory. It is based on the diffeomorphism of the two-dimensional sphere S 2 onto the space of states ℂP 1 and the isomorphism between the groups Pℳ(2) and SO 3 (ℝ). In particular, rotational invariance leads to the appearance of the spin1/2 in a natural way.

After defining, for each many-sorted signature Σ = (S, Σ), the category Ter(Σ), of generalized terms for Σ (which is the dual of the Kleisli category for $${\mathbb {T}_{\bf \Sigma}}$$, the monad in Set S determined by the adjunction $${{\bf T}_{\bf \Sigma} \dashv {\rm G}_{\bf \Sigma}}$$ from Set S to Alg(Σ), the category of Σ-algebras), we assign, to a signature morphism d from Σ to Λ, the functor $${{\bf d}_\diamond}$$ from Ter(Σ) to Ter(Λ). Once defined the mappings that assign, respectively, (...) to a many-sorted signature the corresponding category of generalized terms and to a signature morphism the functor between the associated categories of generalized terms, we state that both mappings are actually the components of a pseudo-functor Ter from Sig to the 2-category Cat. Next we prove that there is a functor TrΣ, of realization of generalized terms as term operations, from Alg(Σ) × Ter(Σ) to Set, that simultaneously formalizes the procedure of realization of generalized terms and its naturalness (by taking into account the variation of the algebras through the homomorphisms between them). We remark that from this fact we will get the invariance of the relation of satisfaction under signature change. Moreover, we prove that, for each signature morphism d from Σ to Λ, there exists a natural isomorphism θ d from the functor $${{{\rm Tr}^{\bf {\bf \Lambda}} \circ ({\rm Id} \times {\bf d}_\diamond)}}$$ to the functor $${{\rm Tr}^{\bf \Sigma} \circ ({\bf d}^* \times {\rm Id})}$$, both from the category Alg(Λ) × Ter(Σ) to the category Set, where d* is the value at d of the arrow mapping of a contravariant functor Alg from Sig to Cat, that shows the invariant character of the procedure of realization of generalized terms under signature change. Finally, we construct the many-sorted term institution by combining adequately the above components (and, in a derived way, the many-sorted specification institution), but for a strict generalization of the standard notion of institution. (shrink)

In this paper, we shall discuss some recent contributions to the project [15, 14, 2, 18, 22, 23] of explaining why no satisfactory system of complete invariants has yet been found for the torsion-free abelian groups of finite rank n ≥ 2. Recall that, up to isomorphism, the torsion-free abelian groups of rank n are exactly the additive subgroups of the n-dimensional vector space ℚn which contain n linearly independent elements. Thus the collection of torsion-free abelian groups of rank (...) at most n can be naturally identified with the set S of all nontrivial additive subgroups of ℚn. In 1937, Baer [4] solved the classification problem for the class Sof rank 1 groups as follows.Let ℙ be the set of primes. If G is a torsion-free abelian group and 0 ≠ x ϵ G, then the p-height of x is defined to behx = sup{n ϵ ℕ ∣ There exists y ϵ G such that pny = x} ϵ ℕ ∪{∞}; and the characteristic χ of x is defined to be the function. (shrink)

We study the expressive power in the finite of the logic Fixed-Point+Counting, the extension of first-order logic which is obtained through adding both the fixed-point constructor and the ability to count. To this end an isomorphism preserving (`generic') model of computation is introduced whose PTime restriction exactly corresponds to this level of expressive power, while its PSpace restriction corresponds to While+Counting. From this model we obtain a normal form which shows a rather clear separation of the relational vs. the (...) arithmetical side of the algorithms involved. In parallel, we study the relations of Fixed-Point+Counting with the infinitary logics L r ∞ω (∃ ≥ m ) m∈ω and the corresponding pebble games. The main result, however, involves the concept of an arithmetical invariant. By this we mean a functor taking every finite relational structure to an expansion of (an initial segment of) the standard arithmetical structure. In particular its values are linearly ordered structures. We establish the existence of a family of arithmetical invariants (J r ) r≥ 1 with the following properties: $\bullet$ The invariants themselves can be evaluated in polynomial time. $\bullet$ A class of finite relational structures is definable in Fixed-Point+Counting if and only if membership can be decided in polynomial time on the basis of the values of one of the invariants. $\bullet$ The invariant J r classifies all finite relational structures exactly up to equivalence with respect to the logic L r ∞ω (∃ ≥ m ) m∈ω . We also give a characterization of Fixed-Point+Counting in terms of sequences of formulae in the L r ωω (∃ ≥ m ) m∈ω : It corresponds exactly to the polynomial time computable families (φ n ) n∈ω in these logics. Towards a positive assessment of the expressive power of Fixed-Point+Counting, it is shown that the natural extension of fixed-point logic by Lindström quantifiers, which capture all the PTime computable properties of cardinalities of definable predicates, is strictly weaker than what we get here. This implies in particular that every extension of fixed-point logic by means of monadic Lindstrom quantifiers, which stays within PTime, must be strictly contained in Fixed-Point+Counting. (shrink)

A structure is called weakly oligomorphic if its endomorphism monoid has only finitely many invariant relations of every arity. The goal of this paper is to show that the notions of homomorphism‐homogeneity, and weak oligomorphy are not only completely analogous to the classical notions of homogeneity and oligomorphy, but are actually closely related. We first prove a Fraïssé‐type theorem for homomorphism‐homogeneous relational structures. We then show that the countable models of the theories of countable weakly oligomorphic structures are mutually homomorphism‐equivalent (...) (we call first order theories with this property weakly ω‐categorical). Furthermore we show that every weakly oligomorphic homomorphism‐homogeneous structure contains (up to isomorphism) a unique homogeneous, homomorphism‐homogeneous core, to which it is homomorphism‐equivalent. As a consequence we obtain that every countable weakly oligomorphic structure is homomorphism‐equivalent to a finite or ω‐categorical structure. As a corollary we obtain a characterization of positive existential theories of weakly oligomorphic structures as the positive existential parts of ω‐categorical theories. (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)