We prove the existence of pairs of models of the same cardinality which are very equivalent according to EF games, but not isomorphic. We continue the paper [4], but we do not rely on it. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim).

Suppose $\lambda$ is a singular cardinal of uncountable cofinality $\kappa$. For a model $\mathscr{M}$ of cardinality $\lambda$, let No ($\mathscr{M}$) denote the number of isomorphism types of models $\mathscr{N}$ of cardinality $\lambda$ which are L$_{\infty\lambda}$- equivalent to $\mathscr{M}$. In [7] Shelah considered inverse $\kappa$- systems $\mathscr{A}$ of abelian groups and their certain kind of quotient limits Gr($\mathscr{A}$)/ Fact($\mathscr{A}$). In particular Shelah proved in [7, Fact 3.10] that for every cardinal $\mu$ there exists an inverse $\kappa$-system $\mathscr{A}$ such that $\mathscr{A}$ consists (...) of abelian groups having cardinality at most $\mu^\kappa$ and card(Gr($\mathscr{A}$)/Fact($\mathscr{A}$)) = $\mu$. Later in [8, Theorem 3.3] Shelah showed a strict connection between inverse $\kappa$-systems and possible values of No (under the assumption that $\theta^\kappa < \lambda$ for every $\theta < \lambda$): if $\mathscr{A}$ is an inverse $\kappa$- system of abelian groups having cardinality < $\lambda$, then there is a model $\mathscr{M}$ such that card $(\mathscr{M}) = \lambda$ and No($\mathscr{M}$) = card(Gr($\mathscr{A}$)/Fact($\mathscr{A}$)). The following was an immediate consequence (when $\theta^\kappa < \lambda$ for every $\theta < \lambda$): for every nonzero $\mu < \lambda$ or $\mu = \lambda^\kappa$ there is a model $\mathscr{M}_\mu$ of cardinality $\lambda$ with No$(\mathscr{M}_\mu) = \mu$. In this paper we show: for every nonzero $\mu \leq \lambda^\kappa$ there is an inverse $\kappa$-system $\mathscr{A}$ of abelian groups having cardinality < $\lambda$ such that card(Gr($\mathscr{A}$)/Fact($\mathscr{A}$)) = $\mu$ (under the assumptions $2^\kappa < \lambda$ and $\theta^{<\kappa} < \lambda$ for all $\theta < \lambda$ when $\mu > \lambda$), with the obvious new consequence concerning the possible value of No. Specifically, the case No($\mathscr{M}$) = $\lambda$ is possible when $\theta^\kappa < \lambda$ for every $\theta < \lambda$. (shrink)

Suppose λ is a singular cardinal of uncountable cofinality κ. For a model M of cardinality λ, let No (M) denote the number of isomorphism types of models N of cardinality λ which are L ∞λ - equivalent to M. In [7] Shelah considered inverse κ- systems A of abelian groups and their certain kind of quotient limits Gr(A)/ Fact(A). In particular Shelah proved in [7, Fact 3.10] that for every cardinal μ there exists an inverse κ-system A such that (...) A consists of abelian groups having cardinality at most μ κ and card(Gr(A)/Fact(A)) = μ. Later in [8, Theorem 3.3] Shelah showed a strict connection between inverse κ-systems and possible values of No (under the assumption that $\theta^\kappa for every $\theta ): if A is an inverse κ- system of abelian groups having cardinality $\theta^\kappa for every $\theta ): for every nonzero $\mu or μ = λ κ there is a model M μ of cardinality λ with No(M μ ) = μ. In this paper we show: for every nonzero μ ≤ λ κ there is an inverse κ-system A of abelian groups having cardinality $2^\kappa and $\theta^{ for all $\theta when $\mu > \lambda$ ), with the obvious new consequence concerning the possible value of No. Specifically, the case No(M) = λ is possible when $\theta^\kappa for every $\theta. (shrink)

In this article we consider linear isomorphisms over the field of rational numbers between the linear spaces ℝ2 and ℝ. We prove that if f is such an isomorphism, then the image by f of the unit disk is a strictly nonmeasurable subset of the real line, which has different properties than classical non-measurable subsets of reals. We shall also consider the question whether all images of bounded measurable subsets of the plane via a such mapping are non-measurable.

The Wittgensteinian thesis that “the result of philosophy is not a number of ‘philosophical propositions,’ but to make propositions clear” has been given various interpretations by subsequent philosophers. Thus, for example, certain British philosophers have confined themselves to the analysis of colloquial language and have developed great skill in sophistical demonstrations intended to show that there is no such thing as a philosophical problem. Another group of philosophers considers language solely as a phenomenon of human behavior and attempts to clarify (...) confusions and misunderstandings within language by utilizing the statistical techniques of contemporary behavior psychology. Still other philosophers, under the impact of logical positivism, have maintained that the clarification of propositions entails the construction of a language with precise rules of formation, transformation, and reference into which one could translate all the non-contradictory portions of ordinary spoken language. (shrink)

Emotion Review, Volume 14, Issue 3, Page 167-181, July 2022. A growing cadre of influential scholars has converged on a circumscribed definition of empathy as restricted only to feeling the same emotion that one perceives another is feeling. We argue that this restrictive isomorphic matching definition is deeply problematic because it deviates dramatically from traditional conceptualizations of empathy and unmoors the construct from generations of scientific research and clinical practice; insistence on an isomorphic form undercuts much of the functional value (...) of empathy from multiple perspectives of analysis; and combining the opposing concepts of isomorphic matching and self-other awareness implicitly requires motivational content, causing the RIM definition to implicitly require the kind of non-matching emotional content that it explicitly seeks to exclude. (shrink)

The purpose of this paper is to elucidate, by means of concepts and theorems drawn from mathematical logic, the conditions under which the existence of a multiverse is a logical necessity in mathematical physics, and the implications of Gödel’s incompleteness theorem for theories of everything.Three conclusions are obtained in the final section: (i) the theory of the structure of our universe might be an undecidable theory, and this constitutes a potential epistemological limit for mathematical physics, but because such a theory (...) must be complete, there is no ontological barrier to the existence of a final theory of everything; (ii) in terms of mathematical logic, there are two different types of multiverse: classes of non-isomorphic but elementarily equivalent models, and classes of model which are both non-isomorphic and elementarily inequivalent; (iii) for a hypothetical theory of everything to have only one possible model, and to thereby negate the possible existence of a multiverse, that theory must be such that it admits only a finite model. (shrink)

We introduce notions of strong and eventual strong non-isolation for types in countable, stable theories. For T superstable or small stable we prove a dichotomy theorem: a regular type over a finite domain is either eventually strongly non-isolated or is non-orthogonal to a NENI type . As an application we obtain the upper bound for Lascar’s rank of a superstable theory which is one-based or trivial, and has fewer than 20 non-isomorphic countable models.

Any "classical" theory of truth will satisfy tarski's criterion ("p" is true if and only if p), And the principle of bivalence (every proposition is either true or false). A non-Classical theory may be obtained by rejecting these principles: - in fact it is shown that rejection of the second entails rejection of the first. If the resulting non-Classical theory is formalized, A system structurally isomorphic to either s4 or s5 is obtained. An attempt is made to show that the (...) essential insights of intuitionist logic may be preserved if we replace their complex and in many respects "unintuitive" propositional logic by a theory of truth which is non-Classical in the sense described above. (shrink)

In his doctor's thesis [1], Henkin has shown that if a formal logic is consistent, and sufficiently complex, then it must admit a non-standard model. In particular, he showed that there must be a model in which that portion of the model which is supposed to represent the positive integers of the formal logic is not in fact isomorphic to the positive integers; indeed it is not even well ordered by what is supposed to be the relation of ≦.For the (...) purposes of the present paper, we do not need a precise definition of what is meant by a standard model of a formal logic. The non-standard models which we shall discuss will be flagrantly non-standard, as for instance a model of the sort whose existence is proved by Henkin. It will suffice if we and our readers are in agreement that a model of a formal logic is not a standard model if either: The relation in the model which represents the equality relation in the formal logic is not the equality relation for objects of the model. That portion of the model which is supposed to represent the positive integers of the formal logic is not well ordered by the relation ≦. That portion of the model which is supposed to represent the ordinal numbers of the formal logic is not well ordered by the relation ≦. (shrink)

We show in the paper that for any non-classifiable countable theory T there are non-isomorphic models and that can be forced to be isomorphic without adding subsets of small cardinality. By making suitable cardinal arithmetic assumptions we can often preserve stationary sets as well. We also study non-structure theorems relative to the Ehrenfeucht-Fraïssé game.

In this paper, we show within RCA 0 that weak Konig's lemma is necessary and sufficient to prove that any (separable) compact group has a Haar measure. Within WKL 0 , a Haar measure is constructed by a non-standard method based on a fact that every countable non-standard model of WKL 0 has a proper initial part isomorphic to itself [10].

George Boolos has described an interpretation of a fragment of ZFC in a consistent second-order theory whose only axiom is a modification of Frege's inconsistent Axiom V. We build on Boolos's interpretation and study the models of a variety of such theories obtained by amending Axiom V in the spirit of a limitation of size principle. After providing a complete structural description of all well-founded models, we turn to the non-well-founded ones. We show how to build models in which foundation (...) fails in prescribed ways. In particular, we obtain models in which every relation is isomorphic to the membership relation on some set as well as models of Aczel's anti-foundation axiom (AFA). We suggest that Fregean extensions provide a natural way to envisage non-well-founded membership. (shrink)

We prove a strengthened version of Shavrukov’s result on the non-isomorphism of diagonalizable algebras of two $\Sigma _1$ -sound theories, based on the improvements previously found by Adamsson. We then obtain several corollaries to the strengthened result by applying it to various pairs of theories and obtain new non-isomorphism examples. In particular, we show that there are no surjective homomorphisms from the algebra $(\mathfrak {L}_T, \Box _T\Box _T)$ onto the algebra $(\mathfrak {L}_T, \Box _T)$. The case of bimodal diagonalizable algebras (...) is also considered. We give several examples of pairs of theories with isomorphic diagonalizable algebras but non-isomorphic bimodal diagonalizable algebras. (shrink)

This paper develops co-ordinated multiple-domain supervenience relations to model determination and dependence relations between complex entities and their constituents by appealing to R-related pairs and by making use of associated isomorphisms. Supervenience relations are devised for order-sensitive and repetition-sensitive mereologies, for mereological systems that make room for many-many composition relations, as well as for hierarchical mereologies that incorporate compositional and hylomorphic structure. Finally, mappings are provided for theories that consider wholes to be prior to their parts.

Sufficient syntactic conditions for canonicity in intermediate and intuitionistic modal logics are given. We present a new technique which does not require semantic first-order reduction and which is constructive in the sense that it works in an intuitionistic metatheory through a model without points which is classically isomorphic to the usual canonical model.

Poincaré's claim that Euclidean and non-Euclidean geometries are translatable has generally been thought to be based on his introduction of a model to prove the consistency of Lobachevskian geometry and to be equivalent to a claim that Euclidean and non-Euclidean geometries are logically isomorphic axiomatic systems. In contrast to the standard view, I argue that Poincaré's translation thesis has a mathematical, rather than a meta-mathematical basis. The mathematical basis of Poincaré's translation thesis is that the underlying manifolds of Euclidean and (...) Lobachevskian geometries are homeomorphic. Assuming as Poincaré does that metric relations are not factual, it follows that we can rewrite a physical theory using Euclidean geometry as one using Lobachevskian geometry and express the same facts. (shrink)

For the first time in history, scholars working on language and culture from within an evolutionary epistemological framework, and thereby emphasizing complementary or deviating theories of the Modern Synthesis, were brought together. Of course there have been excellent conferences on Evolutionary Epistemology in the past, as well as numerous conferences on the topics of Language and Culture. However, until now these disciplines had not been brought together into one all-encompassing conference. Moreover, previously there never had been such stress on alternative (...) and complementary theories of the Modern Synthesis. Today we know that natural selection and evolution are far from synonymous and that they do not explain isomorphic phenomena in the world. ‘Taking Darwin seriously’ is the way to go, but today the time has come to take alternative and complementary theories that developed after the Modern Synthesis, equally seriously, and, furthermore, to examine how language and culture can merit from these diverse disciplines. As this volume will make clear, a specific inter- and transdisciplinary approach is one of the next crucial steps that needs to be taken, if we ever want to unravel the secrets of phenomena such as language and culture. (shrink)

Let $\scr{M}=\langle M,+,<,0,S\rangle $ be a linear o-minimal expansion of an ordered group, and $G=\langle G,\oplus ,e_{G}\rangle $ an n-dimensional group definable in $\scr{M}$ . We show that if G is definably connected with respect to the t-topology, then it is definably isomorphic to a definable quotient group U/L, for some convex ${\ssf V}\text{-definable}$ subgroup U of $\langle M^{n},+\rangle $ and a lattice L of rank equal to the dimension of the 'compact part' of G.

QUALITY AND QUANTITY are fairly generally regarded as complementary terms, although the profound reason for their comple- mentarism is often far from being understood, this reason lying in the 'polar' correspondence referred to toward ...

The Reign of Quantity gives a concise but comprehensive view of the present state of affairs in the world, as it appears from the point of view of the 'ancient wisdom', formerly common both to the East and to the West, but now almost entirely lost sight of. The author indicates with his fabled clarity and directness the precise nature of the modern deviation, and devotes special attention to the development of modern philosophy and science, and to the part played (...) by them, with their accompanying notions of progress and evolution, in the formation of the industrial and democratic society which we now regard as 'normal'. Guénon sees history as a descent from Form (or Quality) toward Matter (or Quantity); but after the Reign of Quantity-modern materialism and the 'rise of the masses'-Guénon predicts a reign of 'inverted quality' just before the end of the age: the triumph of the 'counter-initiation', the kingdom of Antichrist. This text is considered the magnum opus among Guénon's texts of civilizational criticism, as is Symbols of Sacred Science among his studies on symbols and cosmology, and Man and His Becoming according to the Vedanta among his more purely metaphysical works. (shrink)

Publié pour la première fois en 1945, ce texte oppose aux religions historiques la tradition unique et originelle de la connaissance ésotérique. Le monde moderne est décrit comme une monstruosité qui ne reconnaît aucune transcendance et privilégie le règne de la contingence, du quantitatif et de l'individualisme.

Description: Contents: Preface 1. General Remarks on the Vedanta 2. Fundamental Distinction Between The Self and the Ego 3. The Vital Centre of the Human Being, Seat of Brahma 4. Purusha and Prakriti 5. Purusha Unaffected by Individual Modifications 6. The Degrees of Individual Manifestation 7. Buddhi or the Higher Intellect 8. Manas or the Inward Sense : The Ten External Faculties of Sensation and Action 9. The Envelopes of the Self ; The Five Vayus or Vital Functions 10. The (...) Essential Unity and Identity of the Self in all the States of the Being 11. The Different Conditions of Atma in the Human Being 12. The Waking State or the Condition of Vaishwanara 13. The Dream State or the Condition of Taijasa 14. The State of Deep Sleep or the Condition of Prajna 15. The Unconditioned State of Atma 16. The Symbolical Representation of Atma and its Conditions by the Sacred Monosyllable Om 17. The Posthumous Evolution of the Human Being 18. The Reabsorption of the Individual Faculties 19. Differences in the Posthumous Conditions According to the Degrees of Knowledge 20. The Coronal Artery and the Solar Ray 21. The Divine Journey of the Being on the Path of Liberation 22. Final Deliverance 23. Videha-mukti and Jivana-mukti 24. The Spiritual State of the Yogi : The Supreme Identity. (shrink)

Buddhist concepts of personal life and responsibilities.

The concluding chapter lays down the essential conditions for any genuine understanding between East and West, which can only come through the work of those who have attained, at least in some degree, to the realization of 'wisdom uncreate' ...

A study of the constitution and development of the human being from the metaphysical point of view, with special reference to Vedantic doctrine.