_Supervision of Sandplay Therapy_, the first book on this subject, is an internationally-based volume that describes the state of the art in supervision of sandplay therapy. Recognizing that practitioners are eager to incorporate sandplay therapy into their practice, Harriet Friedman and Rie Rogers Mitchell respond to the need for new information, and successfully translate the theories of sandplay therapy into supervision practice. The book provides a meaningful connection and balance between theoretical principles, practical application, and ongoing therapeutic encounter (...) involved in sandplay. Divided into six sections, contributors cover: original supervision models contemporary supervision models special challenges in supervision international sandplay supervision supervision of special groups connections with other arts therapies. _Supervision of Sandplay Therapy _expands the vision of what is possible in supervision and will be vital reading for those studying supervision and sandplay therapy, as well as for those wanting to provide a depth-oriented approach during supervision. (shrink)

Sandplay is one of the fastest growing therapies. What are its origins, who were it pioneers, and how have they influenced the current practice of sandplay? What does the future hold? Rie Rogers Mitchell and Harriet S. Friedman have written a unique book that answers all these questions and many more. They give an overview of the historical origins of sandplay, including biographical profiles of the innovators together with discussions of their seminal writings. The five main therapeutic trends (...) are explored, and in a final chapter the future of sandplay is discussed through addressing emerging issues and concerns. A special feature is a comprehensive international bibliography as well as a listing of sandtray videotapes and audiotapes. (shrink)

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)

The equiconsistency of a measurable cardinal with Mitchell order o=κ++ with a measurable cardinal such that 2κ=κ++ follows from the results by W. Mitchell [13] and M. Gitik [7]. These results were later generalized to measurable cardinals with 2κ larger than κ++ .In Friedman and Honzik [5], we formulated and proved Eastonʼs theorem [4] in a large cardinal setting, using slightly stronger hypotheses than the lower bounds identified by Mitchell and Gitik , for a suitable μ, (...) instead of the cardinals with the appropriate Mitchell order).In this paper, we use a new idea which allows us to carry out the constructions in Friedman and Honzik [5] from the optimal hypotheses. It follows that the lower bounds identified by Mitchell and Gitik are optimal also with regard to the general behavior of the continuum function on regulars in the context of measurable cardinals. (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)

We continue the study of adequate sets which we began in (Krueger in Forcing with adequate sets of models as side conditions) by introducing the idea of a strongly adequate set, which has an additional requirement on the overlap of two models past their comparison point. We present a forcing poset for adding a club to a fat stationary subset of ω 2 with finite conditions, thereby showing that a version of the forcing posets of Friedman (Set theory: Centre (...) de Recerca Matemàtica, Barcelona, 2003–2004, Trends in Mathematics. Birkhäuser Verlag, 2006) and Mitchell (Trans Am Math Soc 361(2):561601, 2009) for adding a club on ω 2 can be developed in the context of adequate sets. (shrink)

Most of the research in foundations of mathematics that I do in some way or another involves the use of nonstandard models. I will give a few examples, and indicate what is involved.

Are humans naturally predisposed to religion and supernatural beliefs? If so, does this naturalness provide a moral foundation for religious freedom? This volume offers a cross-disciplinary approach to these questions, engaging in a range of contemporary debates at the intersection of religion, cognitive science, sociology, anthropology, political science, epistemology, and moral philosophy. The contributors to this original and important volume present individual, sometimes opposing points of view on the naturalness of religion thesis and its implications for religious freedom. Topics include (...) the epistemological foundations of religion, the relationship between religion and health, and a discussion of the philosophical foundations of religious freedom as a natural, universal right, drawing implications for the normative role of religion in public life. By challenging dominant intellectual paradigms, such as the secularization thesis and the Enlightenment view of religion, the volume opens the door to a powerful and provocative reconceptualization of religious freedom. (shrink)

oris, 1986, 17-34, Seminarberichte 86, Humboldt University, Berlin, where, with G. Cherlin, we gave a detailed proof of a result of Alexei L. Semenov that the theory of (N, +, 2x) is decidable and admits quantifier elimination in an expansion of the language containing the Presburger..

“Sentential reflection” in the sense of [Fr03] is based on reflecting down from a category of classes. “Elemental sentential reflection” is based on reflecting down from a category of elemental classes. We present various forms of elemental sentential reflection, which are shown to interpret and be interpretable in certain set theories with large cardinal axioms.

We present two forms of “sentential reflection”, which are shown to be mutually interpretable with Z2 and ZFC, respectively.

Gödel's legacy is still very much in evidence. We will not attempt to properly discuss the full impact of his work and all of the ongoing important research programs that it suggests. This would require a book length manuscript. Indeed, there are several books discussing the Gödel legacy from many points of view, including, for example, (Wang 1987, 1996), (Dawson 2005), and the historically comprehensive five volume set (Gödel 1986-2003).

We say that E is R-sparse if f(Ek) has no interior, for each k 2 N and f : Rk ! R de nable in R. (Throughout, \de nable" means \de nable without parameters".) In this note, we consider the extent to which basic metric and topological properties of subsets of R de nable in (R;E)# are determined by the corresponding properties of subsets of R de nable in (R;E), when R is an o-minimal expansion of (R;<;+;0;1) and E is (...) R-sparse. The precise statement of the main result is a bit complicated, but we can state some special cases now. (shrink)

A review of Drew Hyland's Questioning Platonism.

Saunders Mac Lane has drawn attention many times, particularly in his book Mathematics: Form and Function, to the system of set theory of which the axioms are Extensionality, Null Set, Pairing, Union, Infinity, Power Set, Restricted Separation, Foundation, and Choice, to which system, afforced by the principle, , of Transitive Containment, we shall refer as . His system is naturally related to systems derived from topos-theoretic notions concerning the category of sets, and is, as Mac Lane emphasises, one that is (...) adequate for much of mathematics. In this paper we show that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke–Platek set theory and even the Axiom of Constructibility to Mac Lane's axioms; our method requires a close study of Axiom H, which was proposed by Mitchell; we digress to apply these methods to subsystems of Zermelo set theory , and obtain an apparently new proof that is not finitely axiomatisable; we study Friedman's strengthening of , and the Forster–Kaye subsystem of , and use forcing over ill-founded models and forcing to establish independence results concerning and ; we show, again using ill-founded models, that proves the consistency of ; turning to systems that are type-theoretic in spirit or in fact, we show by arguments of Coret and Boffa that proves a weak form of Stratified Collection, and that is a conservative extension of for stratified sentences, from which we deduce that proves a strong stratified version of ; we analyse the known equiconsistency of with the simple theory of types and give Lake's proof that an instance of Mathematical Induction is unprovable in Mac Lane's system; we study a simple set theoretic assertion—namely that there exists an infinite set of infinite sets, no two of which have the same cardinal—and use it to establish the failure of the full schema of Stratified Collection in ; and we determine the point of failure of various other schemata in . The paper closes with some philosophical remarks. (shrink)

by Harvey M. Friedman Department of Mathematics Ohio State University friedman@math.ohio-state.edu www.math.ohio-state.edu/~friedman/ April 5, 1997..

The use of x[y,z,w] rather than the more usual y Πx has many advantages for this work. One of them is that we have found a convenient way to eliminate any need for axiom schemes. All axioms considered are single sentences with clear meaning. (In one case only, the axiom is a conjunction of a manageable finite number of sentences).

Harold Noonan has recently argued (2003) that one of Lewis’s (1983: 76– 77) arguments for the view that objects persist by perduring is flawed. Lewis’s argument can be divided into two main sections, the first of which attempts to show that it is possible that there exists a world of temporal parts or stages, and the second, which attempts to show that our world is such a world. Noonan claims that there is a flaw in each of these two stages.We (...) argue to the contrary. (shrink)

We present an approach to forcing with finite sequences of models that uses models of two types. This approach builds on earlier work of Friedman and Mitchell on forcing to add clubs in cardinals larger than $\aleph_{1}$, with finite conditions. We use the two-type approach to give a new proof of the consistency of the proper forcing axiom. The new proof uses a finite support forcing, as opposed to the countable support iteration in the standard proof. The distinction (...) is important since a proof using finite supports is more amenable to generalizations to cardinals greater than $\aleph_{1}$. (shrink)

The issue of whether there are laws in biology and the “special science”1 has been of interest owing to the debate about whether scientific explanation requires laws. A well-warn argument goes thus: no laws in social science, no explanations, or at least no scientific explanations, at most explanation-sketches. The conclusion is not just a matter of labeling. If explanations are not scientific they are not epistemically or practically reliable. There are at least three well-known diagnoses of where this argument goes (...) wrong. First, the argument that there are no laws in social science adopts an account of laws that is too stringent, one that not even the physical sciences satisfy (Cartwright 1983, Mitchell 2000). On a less stringent definition, there are plenty of laws in social science (and biology). These laws are, sensu Fodor, “non-strict,” as opposed to the “strict laws” (if any—vide Cartwright 1983) of physics. Second, scientific explanation does not require laws, and when laws do explain, they do so because they satisfy some other requirement on scientific explanation, for example unification, or the identification of causal difference-makers (Friedman 1974, Kitcher 1989, Salmon 1984, Strevens 2009). A third view, increasingly attractive among philosophers of social science and biology is due to James Woodward (2000, 2003). This view, like the second one eschews laws and identifies causes as difference makers. On this view explanations do require regularities, but these regularities need only satisfy a requirement of “invariance” under certain specified circumstances, in order to be explanatory, and.. (shrink)

The content ranking system of reddit.com, the English language Internet’s most popular social news website, plays a large but often unnoticed role in shaping what users see and how they think. By pairing informational cascade theory with textual analysis, I argue that the “karma” system elevates particular forms of content over others and generates numerical cues that unconsciously guide users’ judgments about said content and about the world. By drawing on Heidegger’s account of modern technology, I argue that the karma (...) system both symptomatizes and engenders an ontological perspective according to which things in general are taken as available, evaluable, and disposable. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Josiah Royce: Pragmatist, Ethicist, Philosopher of Religion ed. by Christoph Seibert and Christian PolkeRobin FriedmanJosiah Royce: Pragmatist, Ethicist, Philosopher of Religion Christoph Seibert and Christian Polke, editors. Mohr Siebeck, 2021.In October 2015, the Warburg Haus, Hamburg, held a conference on the American philosopher Josiah Royce that brought together German and American scholars. The papers given at the conference led to this new book, Josiah Royce: Pragmatist, Ethicist, Philosopher (...) of Religion. Edited by the German scholars Christoph Seibert and Christian Polke, the volume includes eleven essays, nine delivered at the 2015 conference, together with additional contributions by Seibert and Polke, the conference organizers.It is valuable to see contemporary German scholars engaging with Royce and to compare their approach with that of American scholars. Americans tend to approach Royce through the tradition of pragmatism in the company of William James, Charles Peirce, and John Dewey, and to minimize the metaphysics in Royce's thought and its idealistic influences. [End Page 116]The book has the goal of developing Royce's thought to show how its insights and concerns may be valuable for contemporary life, not only for scholars of differing philosophical traditions. In their Introduction, Seibert and Polke describe Royce as a "pragmatist of his own kind" (3) and identify three themes in his philosophy. The first is community, which was central to Royce from the beginning and, following his serious study of Peirce, "became the ultimate principle of both his metaphysics and his philosophy of life" (3). The second theme is religion, which, for Royce, involved the "search for the ultimate meaning of his own life, as well as that of his fellows' lives" (3). The third broad theme in Royce is ethics and a lifelong concern with the nature of the examined life. Royce developed his ethical philosophy from an original critique of moral skepticism to the "never-ending task of building communities within and through mankind by way of a critical spirit of true and mutual loyalty" (3). Among these three, the editors consider religion as the most important. They state: "[R]eligion is a very important theme of Royce's philosophy: one may even assume that it is the central subject, which implies that his philosophy may be properly understood only if it is regarded as a philosophy of religion" (8). The centrality of religion is apparent in the essays. Consideration here must be brief.The book's first section, "Bridging Two Continents: Royce between Pragmatism and Idealism," includes three broad-ranging essays that work toward a synthesis between pragmatism and idealism in the thought of Royce. Ludwig Nagl's contribution, "Toward a Global Philosophical Discourse on Religion," discusses Royce's posthumously published "Lectures on Modern Idealism." Nagl uses this work, together with Royce's adaption of Peirce, to help create a bridge between pragmatism and idealism. He further explores Royce's interest in Eastern religion to suggest how Royce's thought might support a broad philosophical study of religion, not restricted to Christianity. Co-editor Seibert's article, "The Ethical Nucleus of Reality," focuses on Royce's epistemology and his ethics, while touching briefly on religion. The essay shows how Royce combined pragmatic and idealistic threads from his earliest writings, with particular attention to his book The Spirit of Modern Philosophy (SM). Royce's SM is also central to the American philosopher Douglas Anderson's essay "What Philosophy Meant to Josiah Royce." In SM, Royce argues that philosophy constitutes a critical reflection on experience that may be practiced by every person, not only by professional philosophers. Anderson, however, also identifies and criticizes a more elitist, professional aspect in Royce's conception of philosophy. For Anderson, Royce's thought had a pragmatist dimension in that philosophy was a "voluntarily chosen attitude toward conducting our lives" (80). Royce's thought also was idealistic [End Page 117] in its recognition of the finite character of human life and in the search for the whole. Anderson writes that, for Royce, "to understand our experiences and to gain some control of our futures, we must see those experiences in their relations to all of what is—no easy task" (80).The book's second section includes... (shrink)

This Book Incorporates Prof. Friedman S Lectures And Discussions That Were A Part Of The Inter-Cultural Dialogue At Many Levels. His Major Contribution Is In Developing An Approach That Is Within The Framework Of The Human Image.

In September 2003, we released the first results of a national community attitude tracking study about child abuse and child protection. At that time, we concluded that as a community, violence against children was tolerated. The community did not understand or appreciate the seriousness, size and cost of child abuse in Australia. There was evidence that child abuse was not viewed as an important challenge facing children in Australia. A second study conducted in 2006 found that nothing much had changed, (...) indeed community engagement with the issue of child abuse may have even deteriorated. A third study in 2010 found that the community actively avoids the problem of child abuse rating it less concerning than high petrol prices. In 2021, 18 years after the first report was published, we have concluded again that child abuse remains out of sight and out of mind as a community concern. This article describes the findings of this fourth iteration of our survey and analyses the implications for ensuring that individuals are more engaged and committed to taking action to preventing child abuse and/or protecting children from violation. (shrink)

I develop a translation procedure between λ-structures, which correspond to premice in the Friedman–Jensen indexing convention on the one hand and s-structures, which are essentially the same as premice in the Mitchell–Steel indexing scheme.

Mitchell, J.C. and E. Moggi, Kripke-style models for typed lambda calculus, Annals of Pure and Applied Logic 51 99–124. The semantics of typed lambda calculus is usually described using Henkin models, consisting of functions over some collection of sets, or concrete cartesian closed categories, which are essentially equivalent. We describe a more general class of Kripke-style models. In categorical terms, our Kripke lambda models are cartesian closed subcategories of the presheaves over a poset. To those familiar with Kripke models (...) of modal or intuitionistic logics, Kripke lambda models are likely to seem adequately ‘semantic’. However, when viewed as cartesian closed categories, they do not have the property variously referred to as concreteness, well- pointedness or having enough points. While the traditional lambda calculus proof system is not complete for Henkin models that may have empty types, we prove strong completeness for Kripke models. In fact, every set of equations that is closed under implication is the theory of a single Kripke model. We also develop some properties of logical relations over Kripke structures, showing that every theory is the theory of a model determined by a Kripke equivalence relation over a Henkin model. We discuss cartesian closed categories but present the main definitions and results without the use of category theory. (shrink)

An ideal in a commutative ring R with unit is a nonempty I Õ R such that for all x,y Œ I, z Œ R, we have x+y and xz Œ I. A set of generators for I is a subset of I such that I is the least ideal containing that subset.

The kind of unknowability I will discuss concerns the count of certain natural finite sets of objects. Even the situation with regard to our present strong formal systems is rather unclear. One can just profitably focus on that, putting aside issues of general unknowability.

We explore Woodin's Universality Theorem and consider to what extent large cardinal properties are transferred into HOD (and other inner models). We also separate the concepts of supercompactness, supercompactness in HOD and being HOD-supercompact. For example, we produce a model where a proper class of supercompact cardinals are not HOD-supercompact but are supercompact in HOD. Additionally we introduce a way to measure the degree of HOD-supercompactness of a supercompact cardinal, and we develop methods to control these degrees simultaneously for a (...) proper class of supercompact cardinals. Finally, we also produce a model in which the unique supercompact cardinal is also the only strongly compact cardinal, no cardinal is supercompact up to an inaccessible cardinal, level by level inequivalence holds and the unique supercompact cardinal is not HOD-supercompact. (shrink)

In Four Seminars, Heidegger reviews the entire trajectory of his thought and offers unique perspectives on fundamental aspects of his work. First published in French in 1976, these seminars were translated into German with Heidegger’s approval and reissued in 1986 as part of his Gesamtausgabe, volume 15. Topics considered include the Greek understanding of presence, the ontological difference, the notion of system in German Idealism, the power of naming, the problem of technology, danger, and the event. Heidegger’s engagements with his (...) philosophical forebears—Parmenides, Heraclitus, Kant, and Hegel—continue in surprising dialogues with his contemporaries—Husserl, Marx, and Wittgenstein. While providing important insights into how Heidegger conducted his lectures, these seminars show him in his maturity reflecting back on his philosophical path. An important text for understanding contemporary philosophical debates, Four Seminars provides extraordinarily rich material for students and scholars of Heidegger. (shrink)

The Borel reducibility theory of Polish equivalence relations, at least in its present form, was initiated independently in [FS89] and [HKL90]. There is now an extensive literature on this topic, including fundamental work on the Glimm-Effros dichotomy in [HKL90], on countable Borel equivalence relations in [DJK94], and on Polish group actions in [BK96].

We present some examples of mathematically natural nonrecursive sets of integers and relations on integers by combining results from Part 1, recursion theory, and from the negative solution to Hilbert’s 10th Problem ([3], [1], and [2]).

It has been accepted since the early part of the Century that there is no problem formalizing mathematics in standard formal systems of axiomatic set theory. Most people feel that they know as much as they ever want to know about how one can reduce natural numbers, integers, rationals, reals, and complex numbers to sets, and prove all of their basic properties. Furthermore, that this can continue through more and more complicated material, and that there is never a real problem.

We discuss several axiomatizations of set theory in first order predicate calculus with epsilon and a constant symbol W, starting with the simple system K(W) which has a strong equivalence with ZF without Foundation. The other systems correspond to various extensions of ZF by certain large cardinal hypotheses. These axiomatizations are unusually simple and uncluttered, and are highly suggestive of underlying philosophical principles that generate higher set theory.

1. Transfer principles from N to On. A. Mahlo cardinals. B. Weakly compact cardinals. C. Ineffable cardinals. D. Ramsey cardinals. E. Ineffably Ramsey cardinals. F. Subtle cardinals. G. From N to (...) 4. Decidability of statements on N. 5. Decidability of statements on shrink)