There have been numerous results showing that a measurable cardinal κ can carry exactly α normal measures in a model of GCH, where a is a cardinal at most κ⁺⁺. Starting with just one measurable cardinal, we have [9] (for α = 1), [10] (for α = κ⁺⁺, the maximum possible) and [1] (for α = κ⁺, after collapsing κ⁺⁺) . In addition, under stronger large cardinal hypotheses, one can handle the remaining cases: [12] (starting with a measurable cardinal of (...) Mitchell order α ) , [2] (as in [12], but where κ is the least measurable cardinal and α is less than κ, starting with a measurable of high Mitchell order) and [11] (as in [12], but where κ is the least measurable cardinal, starting with an assumption weaker than a measurable cardinal of Mitchell order 2). In this article we treat all cases by a uniform argument, starting with only one measurable cardinal and applying a cofinality-preserving forcing. The proof uses κ-Sacks forcing and the "tuning fork" technique of [8]. In addition, we explore the possibilities for the number of normal measures on a cardinal at which the GCH fails. (shrink)

Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at${\kappa ^{ + + }}$, assuming that$\kappa = {\kappa ^{ < \kappa }}$and there is a weakly compact cardinal aboveκ.If in additionκis supercompact then we can forceκto be${\aleph _\omega }$in the extension. The proofs combine the techniques of adding and then destroying (...) a nonreflecting stationary set or a${\kappa ^{ + + }}$-Souslin tree, variants of Mitchell’s forcing to obtain the tree property, together with the Prikry-collapse poset for turning a large cardinal into${\aleph _\omega }$. (shrink)

__The View from Within_ _examines the character of reason and the ability of an individual to effectively distance himself from the normative framework in which he functions in order to be self-critical and innovative. To accomplish this task, Menachem Fisch and Yitzhak Benbaji critically employ or reject the recent writings of Brandom, Friedman, Frankfurt, Walzer, Davidson, Williams, Habermas, Rorty, and McDowell to offer a fundamental analysis of the character of reason and the problem of relativism. This ambitious book (...) forcefully raises the problem of rational normative change and makes the unique and insightful claim that although we cannot be _convinced _by normative criticism to modify or replace our norms, we can be rationally _motivated_ to do so by the effect of exposure to trusted critics. Its unprecedented analysis, with its solution to the problem of normative self-criticism that has baffled philosophers for the past sixty years, will be welcomed by both students and scholars of philosophy. “__The View from Within_ _is a thorough evaluation of the arguments made by contemporary philosophers about the normative character of reason and the derivative problem of relativism. Fisch and Benbaji have admirably compared and contrasted competing positions, and with a balanced critique, they have made a sustained effort to ‘save’ rationality and provide new guideposts for its philosophical evaluation. A timely and important contribution.” —_Alfred I. Tauber, Boston University_. (shrink)

Given a cardinal κ that is λ-supercompact for some regular cardinal λ⩾κ and assuming GCH, we show that one can force the continuum function to agree with any function F:[κ,λ]∩REG→CARD satisfying ∀α,β∈domα F. Our argument extends Woodinʼs technique of surgically modifying a generic filter to a new case: Woodinʼs key lemma applies when modifications are done on the range of j, whereas our argument uses a new key lemma to handle modifications done off of the range of j on the (...) ghost coordinates. This work answers a question of Friedman and Honzik [5]. We also discuss several related open questions. (shrink)

Women have historically been prevented from living autonomously by systematic injustice, subordination, and oppression. The lingering effects of these practices have prompted many feminists to view autonomy with suspicion. Here, Marilyn Friedman defends the ideal of feminist autonomy. In her eyes, behavior is autonomous if it accords with the wants, cares, values, or commitments that the actor has reaffirmed and is able to sustain in the face of opposition. By her account, autonomy is socially grounded yet also individualizing and (...) sometimes socially disruptive, qualities that can be ultimately advantageous for women. Friedman applies the concept of autonomy to domains of special interest to women. She defends the importance of autonomy in romantic love, considers how social institutions should respond to women who choose to remain in abusive relationships, and argues that liberal societies should tolerate minority cultural practices that violate women's rights so long as the women in question have chosen autonomously to live according to those practices. (shrink)

Martin Buber: The Life of Dialogue , the first study in any language to provide a complete overview of Buber's thought, remains the definitive guide to the full range of his work and the starting point for all modern Buber scholarship. As well as summarizing Buber's early intellectual development and attitudes - his mysticism, his youthful existentialism, his philosophy of Judaism and religious socialism - it focuses on the two crucial issues of his mature thought: his dialogic or I-Thou philosophy, (...) and his probing of the nature and redemption of evil. As a sensitive, intuitive and perennially fascinating account of one of the twentieth century's great spiritual teachers, and as an influential classic in its own right, Martin Buber: The Life of Dialogue reveals the implications of Buber's thought for theory of knowledge, education, philosophy, myth, history and Judaic and Christian belief. This fully revised and expanded 4th edition includes a new preface from the author, an expanded bibliography incorporating new Buber scholarship, and two new appendices in the form of essays on Buber's influence on Emmanuel Levinas and Mikhail Bakhtin. (shrink)

French theorists have recently proposed a theory which describes all literature in terms of the probable, the vraisemblable.6 This poetics of the probable commences with a purely relativistic claim. What is probable not only changes in accordance with the audience’s concept of reality but also changes in accordance with the needs of the story and with the narrative possibilities open to various genres. It includes all of the norms and models making a given text understandable to the reader, however outlandish (...) and eccentric it may be. Various levels of the vraisemblable are distinguished from each other, and the vraisemblable based upon models of the world or the world view prevalent in a given culture is scrupulously separated from types of the vraisemblable that are based upon literary, generic models. However, as Tzvetan Todorov gladly admits, “the two notions tend to melt into each other.”7 He and other theorists see the literary genre itself as one of the models “probabilizing” a given text, at times in direct contradiction with “natural” models of the world. Still others see the world, or the prevalent world view, as the universal text which probabilizes other texts.[…]Rather than how this new concept of the vraisemblable, of probability, fares in relation to farce in films, theater, and literature, I prefer to ask how farce fares in relation to it. I think that it doesn’t do well. Farce shows that it is a mistake to unify the possibilities for understanding a text and the possibilities for understanding its fictional world within a single integrated concept. It is a further mistake to term such a concept “probability.” When dealing with the fantastic or the marvelous, for instance, probabilizing the text and probabilizing the work’s fictional world may legitimately be unified under a single principle or concept. If we agree to change or suspend a fundamental belief about the world, we will be able to perceive as logical and probable not only the texts but also the fictional worlds of such works. Coleridge’s “willing suspension of disbelief” excellently fits works that are written in these genres. Such works require that we temporarily forget a certain general belief about the world. After we’ve agreed, as a result of this suspension, to include certain elements in the world , we may then rebuild the fictional world while using the rest of our beliefs along with this new assumption. Farce’s incredibility creates a different situation. Before we have even begun to recover from an attack upon one probability principle, another is undermined. In farce the rules of probability are not neutralized in one specific realm of reality; their inversion operates constantly on all of reality’s realms. The image of a seemingly real world, reminding us of our own world and of our own set of probabilities, is all we need to be constantly astonished by farce’s persistently novel deviations from predictable probability. Here, in direct opposition to the marvelous, the new deviations must keep on coming from unexpected directions, because farce won’t allow its incredibilities to consolidate and become new rules of probability. The fictional world of overt farce lacks a specific factor, the one that would have probabilized all of the other factors of this world. Such an Archimedean center might seem to be located in the genre’s definition—this genre, however, does not probabilize the incredible. Farce’s definition does not make farce’s world more intelligible in any sense of that word. At the very most, farce allows us to forgo such intelligibility without transgressing the established boundaries of art.6. For an excellent presentation of their views, see Jonathan Culler, Structuralist Poetics: Structuralism, Linguistics, and the Study of Literature , pp. 131-60.7. Tzvetan Todorov, The Poetics of Prose, trans. Richard Howard , p. 83. See also his introduction to Communications 11 : 2. (shrink)

How can metaphors account for the formation of mathematical concepts, for changes in mathematical practices, or for the handling of mathematical problems? Following Hans Blumenberg’s thought, this paper aims to unfold a possible answer to these questions by viewing the metaphorical frameworks accompanying these changes as essential for an understanding of how changes in mathematical practices have been accounted for. I will focus especially on cases in which these changes were caused by encounters with a mathematical object which did not (...) yet have a well-defined concept, but also show that such indeterminacy remains with the mathematical concept even after it is considered ‘well-defined’. As the paper will show, this ‘forefield’ [Vorfeld] of the concept is addressed by Blumenberg’s account of metaphorology on the one hand, and accompanied by his later account of nonconceptuality [Unbegrifflichkeit] on the other hand. While Blumenberg himself did not develop a full-fledged philosophy of mathematics or of mathematical practices, I aim to show that one can nevertheless extract from his writings a unique position concerning the role metaphors play in mathematics. To this end, Blumenberg’s account of nautical and oceanic metaphors and Alexandre Grothendieck’s philosophy of mathematical practice provide fruitful starting points. (shrink)

Toldot ha-meḥaber - Petaḥ davar -- Psikhonoyroʼimunologyah, yaḥase gomlin nefesh guf -- Hakarat ha-ṭov la-tseṿet ha-metapel -- Peʻilut hatsalah ha-ʻalulah le-hasev nezaḳim -- ha-Ḥoleh ha-sofani ha-noṭeh la-mut -- ʻAl saf gesher ha-ḥayim -- Ḳedushat ha-ḥayim -- Ḥoleh dimenṭi, ḥasar deʻah, Altshaimer -- Ḥolim ḳashim ʻim ekhut ḥayim yerudah u-mugbaluyot -- Refuʼat nashim (geniḳologyah), ʻubarim, yiludim ṿi-yeladim -- Bediḳot seḳer -- Amirat emet la-ḥoleh.

Assuming some large cardinals, a model of ZFC is obtained in which $\aleph_{\omega+1}$ carries no Aronszajn trees. It is also shown that if $\lambda$ is a singular limit of strongly compact cardinals, then $\lambda^+$ carries no Aronszajn trees.

Our theme is that not every interesting question in set theory is independent of ZFC. We give an example of a first order theory T with countable D(T) which cannot have a universal model at ℵ1 without CH; we prove in ZFC a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove--again in ZFC--that for a large class of cardinals there is no universal linear order (e.g. in every regular $\aleph_1 < (...) \lambda < 2^{\aleph_0}$). In fact, what we show is that if there is a universal linear order at a regular λ and its existence is not a result of a trivial cardinal arithmetical reason, then λ "resembles" ℵ1--a cardinal for which the consistency of having a universal order is known. As for singular cardinals, we show that for many singular cardinals, if they are not strong limits then they have no universal linear order. As a result of the nonexistence of a universal linear order, we show the nonexistence of universal models for all theories possessing the strict order property (for example, ordered fields and groups, Boolean algebras, p-adic rings and fields, partial orders, models of PA and so on). (shrink)

We prove that the statement "For every pair A, B, stationary subsets of ω 2 , composed of points of cofinality ω, there exists an ordinal α such that both A ∩ α and $B \bigcap \alpha$ are stationary subsets of α" is equiconsistent with the existence of weakly compact cardinal. (This completes results of Baumgartner and Harrington and Shelah.) We also prove, assuming the existence of infinitely many supercompact cardinals, the statement "Every stationary subset of ω ω + 1 (...) has a stationary initial segment.". (shrink)

This is a survey paper giving a self-contained account of Shelah's theory of the pcf function pcf={cf:D is an ultrafilter on a}, where a is a set of regular cardinals such that a

Logic and Philosophy of Logic

Logic and Philosophy of Logic, Miscellaneous in Logic and Philosophy of Logic

Model Theory in Logic and Philosophy of Logic

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there are mathematical statements that cannot be proved or refuted using the usual axioms and rules of inference of mathematics.

Kojman, M. and S. Shelah, The universality spectrum of stable unsuperstable theories, Annals of Pure and Applied Logic 58 57–72. It is shown that if T is stable unsuperstable, and 1 [brvbar]T[brvbar], T stable and κ<κ then there is a universal tree of height κ + 1 in cardinality λ.

An interesting question arises in the context of the typically medieval description of the universe presented at the beginning of Maimonides' great law code, the Mishneh Torah. What was Maimonides' own attitude towards that account? Was it meant only as a statement of the best description of nature available at the time matters which make up the bulk of the Mishneh Torah) or was it meant to be a description of the true nature of the universe as it really is, (...) not subject to revision in the light of new paradigms or new models ? Answering this question will lead us to a better understanding of Maimonides' understanding of the nature of science and of what I shall call, for lack of a better term, scientific progress. Maimonides will be shown to hold that while sublunar science can reach perfection and completion such is not possible for superlunar science and that to the extent that the scientific matters in the Mishneh Torah deal with the latter they could not have been presented as the final description of the universe as it truly is. (shrink)

This book explores the emergence of the fundamental political concepts of medieval Jewish thought, arguing that alongside the well known theocratic elements of the Bible there exists a vital tradition that conceives of politics as a necessary and legitimate domain of worldly activity that preceded religious law in the ordering of society. Since the Enlightenment, the separation of religion and state has been a central theme in Western political history and thought, a separation that upholds the freedom of conscience of (...) the individual. In medieval political thought, however, the doctrine of the separation of religion and state played a much different role. On the one hand, it served to maintain the integrity of religious law versus the monarch, whether canon law, Islamic law, or Jewish law. On the other hand, it upheld the autonomy of the monarch and the autonomy of human political agency against theocratic claims of divine sovereignty and clerical authority. Postulating the realm of secular politics leads the author to construct a theory of the precedence of politics over religious law in the organization of social life. He argues that the attempts of medieval philosophers to understand religion and the polity provide new perspectives on the viability of an accommodation between revelation and legislation, the holy and the profane, the divine and the temporal. The book shows that in spite of the long exile of the Jewish people, there is, unquestionably, a tradition of Jewish political discourse based on the canonical sources of Jewish law. In addition to providing a fresh analysis of Maimonides, it analyzes works of Nahmanides, Solomon ibn Adret, and Nissim Gerondi that are largely unknown to the English-speaking reader. Finally, it suggests that the historical corpus of Jewish political writing remains vital today, with much to contribute to the ongoing debates over church-state relations and theocratic societies. (shrink)

For many, the two key thinkers about science in the twentieth century are Thomas Kuhn and Karl Popper, and one of the key questions in contemplating science is how to make sense of theory change. In Creatively Undecided, philosopher Menachem Fisch defends a new way to make sense of the rationality of scientific revolutions. He argues, loosely following Kuhn, for a strong notion of the framework dependency of all scientific practice, while at the same time he shows how such (...) frameworks can be deemed the possible outcomes of keen rational deliberation along Popperian lines. Fisch's innovation is to call attention to the importance of ambiguity and indecision in scientific change and advancement. Specifically, he backs the problem up, looking not at how we might communicate rationally across an already existing divide but at the rational incentive to create an alternative framework in the first place. Creatively Undecided will be essential reading for philosophers of science, and its vivid case study in Victorian mathematics will draw in historians. (shrink)

Starting from cardinals κ κ is measurable, we construct a model for the theory “ZF + n < ω[DCn] + ω + 1 is a measurable cardinal”. This is the maximum amount of dependent choice consistent with the measurability of ω + 1, and by a theorem of Shelah using p.c.f. theory, is the best result of this sort possible.

We prove thatµ=µ <µ , 2 µ =µ + and “there is a non-reflecting stationary subset ofµ + composed of ordinals of cofinality <μ” imply that there is a μ-complete Souslin tree onµ +.

A vast and interesting family of natural semantics for belief revision is defined. Suppose one is given a distance d between any two models. One may then define the revision of a theory K by a formula α as the theory defined by the set of all those models of α that are closest, by d, to the set of models of K. This family is characterized by a set of rationality postulates that extends the AGM postulates. The new postulates (...) describe properties of iterated revisions. (shrink)

In this paper, I defend Rudolf Carnap's Principle of Tolerance from an accusation, due to Michael Friedman, that it is self-defeating by prejudicing any debate towards the logically stronger theory. In particular, Friedman attempts to show that Carnap's reconstruction of the debate between classicists and intuitionists over the foundations of mathematics in his book The Logical Syntax of Language, is biased towards the classical standpoint since the metalanguage he constructs to adjudicate between the rival positions is fully classical. (...) I argue that this criticism is mistaken on two counts: (1) it fails to fully appreciate the freedom with regard to the construction of linguistic frameworks that Carnap intended his Principle to embody, and (2) Friedman's objection underestimates the extent to which the evaluation of a framework is task-relative. I conclude that Tolerance is not self-undermining in the way that Friedman claims it is. While this is a restricted conclusion -- and is not a vindication of Carnap's views on logic and mathematics tout court -- it nonetheless suggests that his tolerant perspective has been dismissed too quickly, even by his supporters. (shrink)

We show that, assuming the consistency of a supercompact cardinal, the first inaccessible cardinal can satisfy a strong form of a Löwenheim–Skolem–Tarski theorem for the equicardinality logic L, a logic introduced in [5] strictly between first order logic and second order logic. On the other hand we show that in the light of present day inner model technology, nothing short of a supercompact cardinal suffices for this result. In particular, we show that the Löwenheim–Skolem–Tarski theorem for the equicardinality logic at (...) κ implies the Singular Cardinals Hypothesis above κ as well as Projective Determinacy. (shrink)

Menachem Lorberbaum; Learning from Mistakes: Resources of Tolerance in the Jewish Tradition, Journal of Philosophy of Education, Volume 29, Issue 2, 30 May 2006.

Menachem Lorberbaum; Learning from Mistakes: Resources of Tolerance in the Jewish Tradition, Journal of Philosophy of Education, Volume 29, Issue 2, 30 May 2006.

The existence of exact upper bounds for increasing sequences of ordinal functions modulo an ideal is discussed. The main theorem gives a necessary and sufficient condition for the existence of an exact upper bound ƒ for a ¦A¦+ is regular: an eub ƒ with lim infI cf ƒ = μ exists if and only if for every regular κ ε the set of flat points in tf of cofinality κ is stationary. Two applications of the main Theorem to set theory (...) are presented. A theorem of Magidor's on covering between models of ZFC is proved using the main theorem : If V-W are transitive models of set theory with ω-covering and GCH holds in V, then κ-covering holds between V and W for all cardinals κ. A new proof of a Theorem by Cummings on collapsing successors of singulars is also given . The appendix to the paper contains a short proof of Shelah's trichotomy theorem, for the reader's convenience. (shrink)

We prove that for every singular cardinal μ of cofinality ω, the complete Boolean algebra contains a complete subalgebra which is isomorphic to the collapse algebra CompCol. Consequently, adding a generic filter to the quotient algebra collapses μ0 to 1. Another corollary is that the Baire number of the space U of all uniform ultrafilters over μ is equal to ω2. The corollaries affirm two conjectures of Balcar and Simon. The proof uses pcf theory.

. The PCF Trichotomy Theorem deals with sequences of ordinal functions on an infinite $\kappa$ modulo some ideal I. If a $<_I$ -increasing sequence of ordinal functions has regular length which is larger than $\kappa^+$ , then by the Trichotomy Theorem the sequence satisfies one of three structural conditions. It was of some interest to find out if the Trichotomy Theorem could hold also for sequences of length $\kappa^+$ . It is shown that this is not the case.

A vast and interesting family of natural semantics for belief revision is defined. Suppose one is given a distance d between any two models. One may then define the revision of a theory K by a formula $\alpha$ as the theory defined by the set of all those models of $\alpha$ that are closest, by d, to the set of models of K. This family is characterized by a set of rationality postulates that extends the AGM postulates. The new postulates (...) describe properties of iterated revisions. (shrink)

_The third of four volumes in a distinguished series, this volume includes chapters on the nature of the communal bond, marriage and family, welfare, taxation, government, and criminal justice_ The four-volume series on the Jewish political tradition that includes this volume seeks to connect the political thought of ancient Israel and the Diaspora with the emerging traditions of the modern Israeli state. The first two volumes dealt with authority and membership, respectively; this third volume, with Madeline Kochen as coeditor, deals (...) with community, with chapters on the communal bond, marriage and family, welfare, taxation, government, and criminal justice. (shrink)

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Discusses the differences between the spies sent by Moses and those sent by Joshua by exploring the spiritual significance of both events and their application today.

heleḳ 1. Be-reshit, Shemot, va-Yikra -- heleḳ 2. ba-Midbar, Devarim.