Suppose that M⊧PA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal M}\models \mathsf{PA}$$\end{document} and X⊆P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak X} \subseteq {\mathcal P}$$\end{document}. If M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal M}$$\end{document} has a finitely generated elementary end extension N≻endM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal N}\succ _\mathsf{end} {\mathcal M}$$\end{document} such that {X∩M:X∈Def}=X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{X \cap M : X \in (...) {{\mathrm{Def}}}\} = {\mathfrak X}$$\end{document}, then there is such an N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal N}$$\end{document} that is, in addition, a minimal extension of M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal M}$$\end{document} iff every subset of M that is Π10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _1^0$$\end{document}-definable in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$$$\end{document} is the countable union of Σ10\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma _1^0$$\end{document}-definable sets. (shrink)

We construct a model of Peano arithmetic in an uncountable language which has no elementary end extension. This answers a question of Gaifman and contrasts with the well-known theorem of MacDowell and Specker which states that every model of Peano arithmetic in a countable language has an elementary end extension. The construction employs forcing in a nonstandard model.

Let U be a well-founded model of ZFC whose class of ordinals has uncountable cofinality, such that U has a Σ n end extension for each n ∈ ω. It is shown in Theorem 1.1 that there is such a model which has no elementary end extension. In the process some interesting facts about topless end extensions (those with no least new ordinal) are uncovered, for example Theorem 2.1: If U is a well-founded model of ZFC, such (...) that U has uncountable cofinality and U has a topless Σ 3 end extension, then U has a topless elementary end extension and also a well-founded elementary end extension, and contains ordinals which are (in U) highly hyperinaccessible. In § 3 related results are proved for κ-like models (κ any regular cardinal) which need not be well founded. As an application a soft proof is given of a theorem of Schmerl on the model-theoretic relation κ → λ. (The author has been informed that Silver had earlier, independently, found a similar unpublished proof of that theorem.) Also, a simpler proof is given of (a generalization of) a characterization by Keisler and Silver of the class of well-founded models which have a Σ n end extension for each n ∈ ω. The case κ = ω 1 is investigated more deeply in § 4, where the problem solved by Theorem 1.1 is considered for non-well-founded models. In Theorems 4.1 and 4.4, ω 1 -like models of ZFC are constructed which have a Σ n end extension for all n ∈ ω but have no elementary end extension. ω 1 -like models of ZFC which have no Σ 3 end extension are produced in Theorem 4.2. The proof uses a notion of satisfaction class, which is also applied in the proof of Theorem 4.6: No model of ZFC has a definable end extension which satisfies ZFC. Finally, Theorem 5.1 generalizes results of Keisler and Morley, and Hutchinson, by asserting that every model of ZFC of countable cofinality has a topless elementary end extension. This contrasts with the rest of the paper, which shows that for well-founded models of uncountable cofinality and for κ-like models with κ regular, topless end extensions are much rarer than blunt end extensions. (shrink)

We study problems of Clote and Paris, concerning the existence of end extensions of models of Σ n -collection. We continue the study of the notion of ‘Γ-fullness’, begun by Wilkie and Paris (Logic, Methodology and Philosophy of Science VIII (Moscow, 1987). Stud. Logic Found. Math., vol. 126, pp. 143–161. North- Holland, Amsterdam, 1989) and introduce and study a generalization of it, to be used in connection with the existence of Σ n -elementary end extensions (instead of plain end (...) extensions). We obtain (a) alternative proofs of results (Adamowicz in Fund. Math. 136, 133–145, 1990) and (Wilkie and Paris in Logic, Methodology and Philosophy of Science VIII (Moscow, 1987). Stud. Logic Found. Math., vol. 126, pp. 143–161. North-Holland, Amsterdam, 1989) related to the problem of Paris and (b) a partial solution to the problem of Clote. (shrink)

This paper concerns intermediate structure lattices Lt, where [MATHEMATICAL SCRIPT CAPITAL N] is an almost minimal elementary end extension of the model ℳ of Peano Arithmetic. For the purposes of this abstract only, let us say that ℳ attains L if L ≅ Lt for some almost minimal elementary end extension of [MATHEMATICAL SCRIPT CAPITAL N]. If T is a completion of PA and L is a finite lattice, then: If some model of T attains L, (...) then every countable model of T does. If some rather classless, ℵ1-saturated model of T attains L, then every model of T does. (shrink)

According to the math tea argument, there must be real numbers that we cannot describe or define, because there are uncountably many real numbers, but only countably many definitions. And yet, the existence of pointwise-definable models of set theory, in which every individual is definable without parameters, challenges this conclusion. In this article, I introduce a flexible new method for constructing pointwise-definable models of arithmetic and set theory, showing furthermore that every countable model of Zermelo-Fraenkel ZF set theory and of (...) Peano arithmetic PA has a pointwise-definable end extension. In the arithmetic case, I use the universal algorithm and its Σ n generalizations to build a progressively elementary tower making any desired individual a n definable at each stage n, while preserving these definitions through to the limit model, which can thus be arranged to be pointwise definable. A similar method works in set theory, and one can moreover achieve V = L in the extension or indeed any other suitable theory holding in an inner model of the original model, thereby fulfilling the resurrection phenomenon. For example, every countable model of ZF with an inner model with a measurable cardinal has an end extension to a pointwise-definable model of ZFC + V = L[μ]. (shrink)

Large cardinals arising from the existence of arbitrarily long end elementary extension chains over models of set theory are studied here. In particular, we show that the large cardinals obtained in this fashion (`unfoldable cardinals') lie in the boundary of the propositions consistent with `V = L' and the existence of 0 ♯ . We also provide an `embedding characterisation' of the unfoldable cardinals and study their preservation and destruction by various forcing constructions.

A model $\mathscr{M} = (M,+,\times, 0,1,<)$ of Peano Arithmetic $({\sf PA})$ is boundedly saturated if and only if it has a saturated elementary end extension $\mathscr{N}$. The ordertypes of boundedly saturated models of $({\sf PA})$ are characterized and the number of models having these ordertypes is determined. Pairs $(\mathscr{N},M)$, where $\mathscr{M} \prec_{\sf end} \mathscr{N} \models({\sf PA})$ for saturated $\mathscr{N}$, and their theories are investigated. One result is: If $\mathscr{N}$ is a $\kappa$-saturated model of $({\sf PA})$ and $\mathscr{M}_0, \mathscr{M}_1 (...) \prec_{\sf end} \mathscr{N}$ are such that $\aleph_1 \leq \mathrm{min}(\mathrm{cf}(M_0),\mathrm{dcf}(M_0)) \leq \mathrm{min}(\mathrm{cf}(M_1), \mathrm{dcf}(M_1)) < \kappa$, then $(\mathscr{N},M_0) \equiv (\mathscr{N},M_1)$. (shrink)

Suppose L = { <,...} is any countable first order language in which < is interpreted as a linear order. Let T be any complete first order theory in the language L such that T has a κ-like model where κ is an inaccessible cardinal. Such T proves the Inaccessibility Scheme. In this paper we study elementary end extensions of models of the inaccessibility scheme.

The existence of End Elementary Extensions of models M of ZFC is related to the ordinal height of M, according to classical results due to Keisler, Morley and Silver. In this paper, we further investigate the connection between the height of M and the existence of End Elementary Extensions of M. In particular, we prove that the theory `ZFC + GCH + there exist measurable cardinals + all inaccessible non weakly compact cardinals are possible heights of models with (...) no End Elementary Extensions' is consistent relative to the theory `ZFC + GCH + there exist measurable cardinals + the weakly compact cardinals are cofinal in ON'. We also provide a simpler coding that destroys GCH but otherwise yields the same result. (shrink)

Keisler in [7] proved that for a strong limit cardinal κ and a singular cardinal λ, the transfer relation κ → λ holds. We analyze the λ -like models produced in the proof of Keisler's transfer theorem when κ is further assumed to be regular. Our main result shows that with this extra assumption, Keisler's proof can be modified to produce a λ -like model M with built-in Skolem functions that satisfies the following two properties: M is generated by a (...) subset C of order-type λ. M can be written as union of an elementary end extension chain 〈Ni: i < δ 〉 such that for each i < δ, there is an initial segment Ci of C with Ci ⊆ Ni, and Ni ∩ = ∅. (shrink)

Wilkie proved in 1977 that every countable model ${\mathcal M}$ of Peano Arithmetic has an elementary end extension ${\mathcal N}$ such that the interstructure lattice $\operatorname {\mathrm {Lt}}({\mathcal N} / {\mathcal M})$ is the pentagon lattice ${\mathbf N}_5$. This theorem implies that every countable nonstandard ${\mathcal M}$ has an elementary cofinal extension ${\mathcal N}$ such that $\operatorname {\mathrm {Lt}}({\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$. It is proved here that whenever ${\mathcal M} \prec {\mathcal N} (...) \models \mathsf {PA}$ and $\operatorname {\mathrm {Lt}}({\mathcal N} / {\mathcal M}) \cong {\mathbf N}_5$, then ${\mathcal N}$ must be either an end or a cofinal extension of ${\mathcal M}$. In contrast, there are ${\mathcal M}^* \prec {\mathcal N}^* \models \mathsf {PA}^*$ such that $\operatorname {\mathrm {Lt}}({\mathcal N}^* / {\mathcal M}^*) \cong {\mathbf N}_5$ and ${\mathcal N}^*$ is neither an end nor a cofinal extension of ${\mathcal M}^*$. (shrink)

A model M = (M, E,...) of Zermelo-Fraenkel set theory ZF is said to be θ-like, where E interprets ∈ and θ is an uncountable cardinal, if |M| = θ but $|\{b \in M: bEa\}| for each a ∈ M. An immediate corollary of the classical theorem of Keisler and Morley on elementary end extensions of models of set theory is that every consistent extension of ZF has an ℵ 1 -like model. Coupled with Chang's two cardinal theorem (...) this implies that if θ is a regular cardinal θ such that $2^{ then every consistent extension of ZF also has a θ + -like model. In particular, in the presence of the continuum hypothesis every consistent extension of ZF has an ℵ 2 -like model. Here we prove: THEOREM A. If θ has the tree property then the following are equivalent for any completion T of ZFC: (i) T has a θ-like model. (ii) $\Phi \subseteq T$ , where Φ is the recursive set of axioms {∃ κ(κ is n-Mahlo and "V κ is a Σ n -elementary submodel of the universe"): n ∈ ω}. (iii) T has a λ-like model for every uncountable cardinal λ. THEOREM B. The following are equiconsistent over ZFC: (i) "There exists an ω-Mahlo cardinal". (ii) "For every finite language L, all ℵ 2 -like models of ZFC(L) satisfy the scheme Φ(L). (shrink)

A model M of PA has the omega-property if it has a subset of order type omega that is coded in an elementary end extension of M. All countable recursively saturated models have the omega-property, but there are also models with the omega-property that are not recursively saturated. The papers is devoted to the study of structural properties of such models.

We give a short proof of a theorem of Kanovei on separating induction and collection schemes for Σ n formulas using families of subsets of countable models of arithmetic coded in elementary end extensions.

We define certain properties of subsets of models of arithmetic related to their codability in end extensions and elementary end extensions. We characterize these properties using some more familiar notions concerning cuts in models of arithmetic.

A model of Peano Arithmetic is short recursively saturated if it realizes all its bounded finitely realized recursive types. Short recursively saturated models of $\PA$ are exactly the elementary initial segments of recursively saturated models of $\PA$. In this paper, we survey and prove results on short recursively saturated models of $\PA$ and their automorphisms. In particular, we investigate a certain subgroup of the automorphism group of such models. This subgroup, denoted $G|_{M(a)}$, contains all the automorphisms of a countable (...) short recursively saturated model of which can be extended to an automorphism of the countable recursively saturated elementary end extension of the model. (shrink)

Proceeding in the theory with extensionality, comprehension for classes, existence of the empty set and the assumption the addition of one element to a set makes again a set we show a week assumption which guarantees existence of a saturated elementary extension of the system of hereditarily finite sets.

Filling the need for an accessible, carefully structured introductory text in symbolic logic, Modern Logic has many features designed to improve students' comprehension of the subject, including a proof system that is the same as the award-winning computer program MacLogic, and a special appendix that shows how to use MacLogic as a teaching aid. There are graded exercises at the end of each chapter--more than 900 in all--with selected answers at the end of the book. Unlike competing texts, Modern Logic (...) gives equal weight to semantics and proof theory and explains their relationship, and develops in detail techniques for symbolizing natural language in first-order logic. After a general introduction featuring the notion of logical form, the book offers sections on classical sentential logic, monadic predicate logic, and full first-order logic with identity. A concluding section deals with extensions of and alternatives to classical logic, including modal logic, intuitionistic logic, and fuzzy logic. For students of philosophy, mathematics, computer science, or linguistics, Modern Logic provides a thorough understanding of basic concepts and a sound basis for more advanced work. (shrink)

We prove a model theoretic generalization of an extension of the Keisler-Morley theorem for countable recursively saturated models of theories having a K-like model, where K is an inaccessible cardinal.

A maximal chain in a tree is called a path, and a tree is called bounded when all its paths contain leaves. This paper concerns itself with first-order theories of bounded trees. We establish some sufficient conditions for the existence of bounded end-extensions that are also partial elementary extensions of a given tree. As an application of tree boundedness, we obtain a conditional axiomatisation of the first-order theory of the class of trees whose paths are all isomorphic to some (...) ordinal \, given the first-order theories of certain classes of bounded trees. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Intersection of Heidegger's Philosophy and His Politics as Reflected in the Views of His Contemporaries at the University of FreiburgRichard DetschThere has been so much speculation in the last ten years or more about the reasons for and the extent of Heidegger's involvement in the Nazi movement that another attempt to come to grips with this important problem might seem superfluous. Amidst the weighty arguments advanced in what (...) has become a multitude of treatises and tomes, however, there has thus far been little interest in dealing with Heidegger's simple statement in his Spiegel interview that he was moved by the "confusion of opinions... from 22 parties... to adopt a political stance that was national and above all social."1 If we suspend our suspicion for the moment that Heidegger is here again manipulating the facts to fit his postwar predicament, as has been amply demonstrated [End Page 407] regarding other details, and examine the evidence we will find that there are indeed indications of the predominance of Heidegger's social concerns with respect to the political situation of the early '30s as well as of his leaning toward an authoritarian form of government an inclination shared, to be sure, by a significantly large number of his compatriots, academic and otherwise. There is one approach in particular toward answering the question of why Heidegger became a Nazi which the present attempt hopes to put in question: the emphasis on his abandonment of his Christian religious origins. The following statement by Theodore Kisiel expresses this line of reasoning most succinctly: "It will take a decade, but it will be not just an anti-Catholic, nor an anti-Christian, but the full-blown anti-religious attitude of the German Geist ripened from German Idealism that sweeps Heidegger into National Socialism, with all the romantic fervor of another religious conversion."2 John Caputo argues, based on the work of Kisiel and van Buren, that the Heidegger of the early twenties was drawing from two "equiprimordial" sources, "the Aristotelian and the Christian." "It was only in the 1930's, in the period of Heidegger's active political engagement with National Socialism, that the twofold root of the tradition was pruned to a single root... a Great Greek beginning from which everything Jewish and Christian, everything Roman, Latin, and Romance, was to be excluded as fallen, derivative, distortive, and inauthentic."3 This shift in perspective, this rejection by Heidegger of earlier Christian impulses in his work led to "fatal" lapses in the ethical realm: "... by leaving behind Kierkegaard and Luther... and by treating Aristotle as a fading echo of the primordially early Greek, Heidegger left the question of Being in a state of utter mystification about ethico-political matters, about the matter of our concrete responsibilities to one another."4 David Farrell Krell makes this lack of respect for and attention to the "other" the centerpiece of an entire study attempting to show Heidegger's propensity for the inhuman ideology of Nazism.5 These arguments derive perhaps from the rhetorical question posed by the Freiburg historian Hugo Ott in 1988: "Didn't this pseudo-religious expectation of salvation [through the mission of the Führer] conceal a monstrous hubris, wasn't the thinking of Heidegger in its radically contained, i.e., in its elementary [End Page 408] simplicity really a surrogate for the Christian view of life thrown overboard?"6 The problem with all such arguments is that "the Christian view of life" did not prevent others of a background similar to Heidegger's from an enthusiastic acceptance of Nazism without an abandonment of their religious faith. In this regard, let the case of Engelbert Krebs, Heidegger's one-time friend and mentor, the Jesuit priest who officiated at his marriage ceremony, serve as an example. Ott deals with Krebs fairly extensively in his Heidegger biography, stressing Krebs's chagrin when Heidegger turned his back on the Catholic Church, but fails to mention at all the enthusiasm for the Nazi movement that Krebs shared with his former friend.During his tenure as dean of the Theological Faculty at the University of Freiburg, Krebs delivered an address in a university lecture... (shrink)

We define and study a new restricted consistency notion RCon ∗ for bounded arithmetic theories T 2 j . It is the strongest ∀ Π 1 b -statement over S 2 1 provable in T 2 j , similar to Con in Krajíček and Pudlák, 29) or RCon in Krajı́ček and Takeuti 107). The advantage of our notion over the others is that RCon ∗ can directly be used to construct models of T 2 j . We apply this by (...) proving preservation theorems for theories of bounded arithmetic of the following well-known kind: The ∀ Π 1 b -separation of bounded arithmetic theories S 2 i from T 2 j is equivalent to the existence of a model of S 2 i which does not have a Δ 0 b -elementary extension to a model of T 2 j . More specific, let M ⊨ Ω 1 nst denote that there is a nonstandard element c in M such that the function n ↦ 2 log c is total in M . Let BLΣ 1 b be the bounded collection schema for Σ 1 b -formulas. We obtain the following preservation results: the ∀ Π 1 b -separation of S 2 i from T 2 j is equivalent to the existence of 1. a model of S 2 i +Ω 1 nst which is 1 b -closed w.r.t. T 2 j , 2. a countable model of S 2 i + BLΣ 1 b without weak end extensions to models of T 2 j. (shrink)

We give alternative proofs of results due to Paris and Wilkie concerning the existence of end extensions of countable models of $B\Sigma_{1}$, that is, the theory of $\Sigma_{1}$ collection.

In this paper, we prove results concerning the existence of proper end extensions of arbitrary models of fragments of Peano arithmetic. In particular, we give alternative proofs that concern a result of Clote :163–170, 1986); :301–302, 1998), on the end extendability of arbitrary models of \-induction, for \, and the fact that every model of \-induction has a proper end extension satisfying \-induction; although this fact was not explicitly stated before, it follows by earlier results of Enayat and Wong (...) and Wong. (shrink)

We show that every model of IΔ0 has an end extension to a model of a theory where log-space computable function are formalizable. We also show the existence of an isomorphism between models of IΔ0 and models of linear arithmetic LA.

We prove that every model of $T = \mathrm{Th}(\omega, countable) has an end extension; and that every countable theory with an infinite order and Skolem functions has 2 ℵ 0 nonisomorphic countable models; and that if every model of T has an end extension, then every |T|-universal model of T has an end extension definable with parameters.

We introduce a principle of local collection for compositional truth predicates and show that it is arithmetically conservative over the classically compositional theory of truth. This axiom states that upon restriction to formulae of any syntactic complexity, the resulting predicate satisfies full collection. In particular, arguments using collection for the truth predicate applied to sentences occurring in any given (code of a) proof do not suffice to show that the conclusion of that proof is true, in stark contrast to the (...) case of the induction scheme. We analyse various further results concerning end-extensions of models of compositional truth and the collection scheme for the compositional truth predicate. (shrink)

We formulate a Π1 sentence τ which is a version of the Tableau consistency of GlΔ0. The sentence τ is true and is provable in GlΔ0 + exp. We construct a model M of GlΔ0+Ω1+τ+BGs1 which has no proper end-extension to a model of GlΔ0+Ω1+τ. Also we prove that GlΔ0+Ω1+τ is not Π1 conservative over GlΔ0+τ.

We consider the quantifier hierarchy of Takahashi [1972] and show how it gives rise to reflection theorems for some large cardinals in ZF, a new natural subtheory of Zermelo's set theory, a potentially useful new reduction of the consistency problem for Quine's NF, and a sharpening of another reduction of this problem due to Boffa.

Every model of IΔ0 is the tally part of a model of the stringlanguage theory Th-FO . We show how to “smoothly” introduce in Th-FO the binary length function, whereby it is possible to make exponential assumptions in models of Th-FO. These considerations entail that every model of IΔ0 + ¬exp is a proper initial segment of a model of Th-FO and that a modicum of bounded collection is true in these models.

. We provide an alternative proof of a theorem of P. Clote concerning end extensions of models of $\Sigma_n$ -induction, for $n \geq 2$.

This book continues from where the authors' previous book, Structural Proof Theory, ended. It presents an extension of the methods of analysis of proofs in pure logic to elementary axiomatic systems and to what is known as philosophical logic. A self-contained brief introduction to the proof theory of pure logic is included that serves both the mathematically and philosophically oriented reader. The method is built up gradually, with examples drawn from theories of order, lattice theory and elementary (...) geometry. The aim is, in each of the examples, to help the reader grasp the combinatorial behaviour of an axiom system, which typically leads to decidability results. The last part presents, as an application and extension of all that precedes it, a proof-theoretical approach to the Kripke semantics of modal and related logics, with a great number of new results, providing essential reading for mathematical and philosophical logicians. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Journal of Aesthetic Education 37.4 (2003) 73-84 [Access article in PDF] Music-Picture:One Form of Synthetic Art Education"Music-picture (a picture drawn through musical perception)" has been widely accepted by art educators in Japan. The purpose of this essay is to propose the making of music-pictures as art education and to put it on afirm theoretical base. I first investigate three gestalt rules: adjacency, continuance, and resemblance, all of which (...) are applicable to the senses of both seeing and hearing. Next I present research on color hearing as one version ofsynaesthesia, which is the comprehensive faculty that binds the five senses in various ways. The well-organized music-picture program by Kaoru Sasaki 1 is introduced as an example. 2 The synthetic art-educational value of music-picture will become clearer through these examinations. The Interrelations between Visual Arts and Music It is exciting that the visual arts meet music in an art class, even if these subjects are entirely different in regard to media, expressive form, the category of perception, and mastery of techniques. Put simply, Picasso's works are visible, whereas Stravinsky's pieces are invisible. Nevertheless, both powerfully convey imaginative and often narrative messages. Relying on intuition alone, I firmly believe both are interchangeable.My research on common methods of organizing both recorded music and painting was my first observation of the mutual relationship between music and visual arts. 3 But what I considered then would not be applied to art class directly because the research was based on my practical experience. It did not have any educational goals. Now I recognize that instructive approaches are necessary to lay the foundation for art classes dealing with interaction between two distinct modalities.In Japan, the Ministry of Education and Science provides the government curriculum guidelines nearly every ten years. It contains descriptions on two subjects: The visual arts and music taught in these classes are at an elementary school level and a junior high school level, both located within a compulsory education system. The new version has been in force since April 2002 (the new school term starts in April in Japan). Objectives in the two educational fields are similar and ideas of what art and music education ought to be are almost the same. The aim common to both in the government curriculum guidelines: "Through expression and appreciation, we [End Page 73] cultivate students' sentiments" invites the linking of music and the visual arts across the curriculum.Many art educators in Japan are interested in this sort of idea and have actually put it into practice. 4 They are engaged in interrelating or even unifying music and the visual arts. Additionally I know other teachers who have designed and proposed many ideas toward this crossover. Listed below are general ideas that seek to combine visual arts with music in an art class.1) Music-picture: to make a pictorial description of impressions received from music.2) Sound map: to make a map of sounds heard at a given location. Each sound is visualized by simple marks like - - -, <<<, ~~~, ooo, and xxx.3) Graphic notation: to write a score which directs the details, whole form, and progression of music by graphics instead of staff notation. Using onomatopoeias or instructions by words are common. It can be improvised, based on intuitively translating visual impacts into sounds.4) Sound toy: to make an original musical instrument of familiar easil obtained materials. For instance, a corrugated cardboard ukulele whose strings are rubber bands, a pair of maracas made of empty cans and soybeans, and so forth.5) Sound sculpture: to make sculpture producing interesting sounds (sometimes noises like creak or clatter). Extensive genres: wood sculpture, metal construction, assemblage (junk arts), kinetic arts, and so on.6) Sound installation: to make a place where participants experience sounds generated by objects or equipment.7) Multimedia: in the broad sense, to mix visuals with music; noises like jingle, tap, and bang! in daily life, or natural environmental sounds like a pit-a-pat of falling rain, a whisper of leaves, or... (shrink)

1. Naturalism Naturalism, it has been said, is the distinctive development in philosophy over the last thirty years. There has been a naturalistic turn away from the a priori methods of traditional philosophy to a conception of philosophy as continuous with natural science. The doctrine has been extensively discussed and has won considerable following in the USA. This is, on the whole, not true of Britain and continental Europe, where the pragmatist tradition never took root, and the temptations of scientism (...) in philosophy were less alluring. Contemporary American naturalism originates in the writings of Quine, the metaphysician of twentieth-century science. With extraordinary panache, he painted a largescale picture of human nature, of language and of the web of belief. I believe that in almost every major respect, it is, like the picture painted by Descartes, the great metaphysician of seventeenth-century science, mistaken. But it evidently appeals to the spirit of the times. So it is worthy of critical examination and careful refutation. I shall argue that the naturalistic turn is a cul-de-sac – a turn that is to be passed by if we are to keep to the highroad of good sense. Naturalism, like so many of Quine’s doctrines, was propounded in response to Carnap. As Quine understood matters, Carnap had been persuaded by Russell’s Our Knowledge of the External World that it is the task of philosophy to demonstrate that our knowledge of the external world is a logical construction out of, and hence can be reduced to, elementary experiences. Quine rejected the reductionism of Carnap’s Logischer Aufbau, and found the idealist basis uncongenial to his own dogmatic realist behaviourism, inspired by Watson and later reinforced by Skinner. The rejection of reductionism and ‘unregenerate realism’, Quine averred, were the sources of his naturalism (FME 72). What exactly was this? We can distinguish in Quine between three different but inter-related programmes for future philosophy: epistemological, ontological and philosophical naturalism. Naturalized epistemology is to displace traditional epistemology, transforming the investigation into ‘an enterprise within natural science’ (NNK 68) – a psychological enterprise of investigating how the ‘input’ of radiation, etc., impinging on the nerve endings of human beings can ‘ultimately’ result in an ‘output’ of our theoretical descriptions of the external world.. (shrink)

An exponential $\exp $ on an ordered field $$. The structure $$ is then called an ordered exponential field. A linearly ordered structure $$ is called o-minimal if every parametrically definable subset of M is a finite union of points and open intervals of M.The main subject of this thesis is the algebraic and model theoretic examination of o-minimal exponential fields $$ whose exponential satisfies the differential equation $\exp ' = \exp $ with initial condition $\exp = 1$. This study (...) is mainly motivated by the Transfer Conjecture, which states as follows:Any o-minimal exponential field $$ whose exponential satisfies the differential equation $\exp ' = \exp $ with initial condition $\exp =1$ is elementarily equivalent to $\mathbb {R}_{\exp }$.Here, $\mathbb {R}_{\exp }$ denotes the real exponential field $$, where $\exp $ denotes the standard exponential $x \mapsto \mathrm {e}^x$ on $\mathbb {R}$. Moreover, elementary equivalence means that any first-order sentence in the language $\mathcal {L}_{\exp } = \{+,-,\cdot,0,1, <,\exp \}$ holds for $$ if and only if it holds for $\mathbb {R}_{\exp }$.The Transfer Conjecture, and thus the study of o-minimal exponential fields, is of particular interest in the light of the decidability of $\mathbb {R}_{\exp }$. To the date, it is not known if $\mathbb {R}_{\exp }$ is decidable, i.e., whether there exists a procedure determining for a given first-order $\mathcal {L}_{\exp }$ -sentence whether it is true or false in $\mathbb {R}_{\exp }$. However, under the assumption of Schanuel’s Conjecture—a famous open conjecture from Transcendental Number Theory—a decision procedure for $\mathbb {R}_{\exp }$ exists. Also a positive answer to the Transfer Conjecture would result in the decidability of $\mathbb {R}_{\exp }$. Thus, we study o-minimal exponential fields with regard to the Transfer Conjecture, Schanuel’s Conjecture, and the decidability question of $\mathbb {R}_{\exp }$.Overall, we shed light on the valuation theoretic invariants of o-minimal exponential fields—the residue field and the value group—with additional induced structure. Moreover, we explore elementary substructures and extensions of o-minimal exponential fields to the maximal ends—the smallest elementary substructures being prime models and the maximal elementary extensions being contained in the surreal numbers. Further, we draw connections to models of Peano Arithmetic, integer parts, density in real closure, definable Henselian valuations, and strongly NIP ordered fields.Parts of this thesis were published in [2–5].Abstract prepared by Lothar Sebastian KrappE-mail: [email protected]: https://d-nb.info/1202012558/34. (shrink)

The "standard account" of Wittgenstein’s relations with the Vienna Circle is that the early Wittgenstein was a principal source and inspiration for the Circle’s positivistic and scientific philosophy, while the later Wittgenstein was deeply opposed to the logical empiricist project of articulating a "scientific conception of the world." However, this telegraphic summary is at best only half-true and at worst deeply misleading. For it prevents us appreciating the fluidity and protean character of their philosophical dialogue. In retrospectively attributing clear-cut positions (...) to Wittgenstein and his interlocutors, it is very easy to read back our current understanding of familiar distinctions into a time when those terms were used in a much more open-ended way. The paper aims to to provide a broader perspective on this debate, starting from the protagonists’ understanding of their respective positions. Too often, the programmatic statements about the nature of their work that are repeated in manifestoes, introductions, and elementary textbooks have occupied center stage in the subsequent secondary literature. Consequently, I focus on a detailed examination of a turning point in their relationship. That turning point is Wittgenstein's charge, in the summer of 1932, that a recently published paper of Carnap's, "Physicalistic Language as the Universal Language of Science", made such extensive and unacknowledged use of Wittgenstein's own ideas that Wittgenstein would, as he put it in a letter to Schlick, "soon be in a situation where my own work shall be considered merely as a reheated version or plagiarism of Carnap’s." While the leading parties in this dispute shared a basic commitment to the primacy of physicalistic language, and the view that all significant languages are translatable, there was a remarkable lack of mutual understanding between them, and deep disagreement about the nature of the doctrines they disputed. Three quarters of a century later, we are so much more conscious of the differences that separated them than the points on which they agreed that it takes an effort of historical reconstruction to appreciate why Wittgenstein once feared that his own work would be regarded as a pale shadow of Carnap’s. (shrink)