This paper introduces the real core model K() and determines the extent of scales in this inner model. K() is an analog of Dodd-Jensen's core model K and contains L(), the smallest inner model of ZF containing the reals R. We define iterable real premice and show that Σ1∩() has the scale property when vR AD. We then prove the following Main Theorem: ZF + AD + V = K() DC. Thus, we obtain the Corollary: If ZF + AD +()L() (...) is consistent, then ZF + AD + DC + α < ω2 -AD is also consistent. (shrink)

This article seeks to explore some issues regarding the different modes of generality at stake in the formation of transdisciplinary concepts within the production of ‘theory’ in the humanities and social sciences. Focused around Jacques Derrida’s seminal account of ‘writing’ in his 1967 book Of Grammatology, the article outlines what it defines as a logic of generalization at stake in Derrida’s elaborations of a quasi-transcendental ‘inscription in general’. Starting out from the questions thereby raised about the relationship between such forms (...) of generality and those historically ascribed to philosophy, the article concludes by contrasting Derrida’s generalized writing with more recent returns to ‘metaphysics’ in the work of Bruno Latour and others. Against the immediately ‘ontological’ orientation of much recent ‘new materialist’ or ‘object-oriented’ thought, the article argues for the necessity of ‘different levels of writing in general’ through a continual folding back of absolute generalization into historically specific disciplinary crossings and exchanges; something suggested by but never really developed in Derrida’s own work. (shrink)

We show, using the fine structure of K, that the theory ZF + AD + X R[X K] implies the existence of an inner model of ZF + AD + DC containing a measurable cardinal above its Θ, the supremum of the ordinals which are the surjective image of R. As a corollary, we show that HODK = K for some P K where K is the Dodd-Jensen Core Model relative to P. In conclusion, we show that the theory ZF (...) + AD + ¬DCR implies that R† exists. (shrink)

Propositional logic -- Predicate logic -- Proof strategies and diagrams -- Mathematical induction -- Set theory -- Functions -- Relations -- Core concepts in abstract algebra -- Core concepts in real analysis.

In this paper, we explore Peirce's work for insights into a theory of learning and cognition for education. Our focus for this exploration is Peirce's paper The Fixation of Belief (FOB), originally published in 1877 in Popular Science Monthly. We begin by examining Peirce's assertion that the study of logic is essential for understanding thought and reasoning. We explicate Peirce's view of the nature of reasoning itself—the characteristic guiding principles or ‘habits of mind’ that underlie acts of inference, the dimensions (...) of and interaction between doubt and belief, and his four methods of resolving or ‘fixing’ belief (i.e., tenacity, authority, a priori, and experimentation). The four methods are then juxtaposed against current models of teaching and learning such as constructivism, schema theory, situated cognition, and inquiry learning. Finally, we discuss Peirce's modes of inference as they relate educationally to the resolution of doubt and beliefs and offer an example of belief resolution from an experienced teacher in a professional development environment. (shrink)

Jensen's celebrated Covering Lemma states that if 0# does not exist, then for any uncountable set of ordinals X, there is a Y∈L such that X⊆Y and |X| = |Y|. Working in ZF + AD alone, we establish the following analog: If ℝ# does not exist, then L(ℝ) and V have exactly the same sets of reals and for any set of ordinals X with |X| ≥ΘL(ℝ), there is a Y∈L(ℝ) such that X⊆Y and |X| = |Y|. Here ℝ is (...) the set of reals and Θ is the supremum of the ordinals which are the surjective image of ℝ. (shrink)

Before one can construct scales of minimal complexity in the Real Core Model, K(R), one needs to develop the fine-structure theory of K(R). In this paper, the fine structure theory of mice, first introduced by Dodd and Jensen, is generalized to that of real mice. A relative criterion for mouse iterability is presented together with two theorems concerning the definability of this criterion. The proof of the first theorem requires only fine structure; whereas, the second theorem applies to real mice (...) satisfying AD and follows from a general definability result obtained by abstracting work of John Steel on L(R). In conclusion, we discuss several consequences of the work presented in this paper relevant to two issues: the complexity of scales in K(R) and the strength of the theory ZF + AD + ¬ DC R. (shrink)

We present a diamond principle ◊R concerning all subsets of Θ, the supremum of the ordinals that are the surjective image of R. We prove that ◊R holds in Steel’s core model K, a canonical inner model for determinacy.

Ethnic competition theory provides a powerful explanation for ethnic conflict, by demonstrating how variation in ethnic mobilization relates to intergroup struggles over scarce resources. However, the tendency to capture such relationships at the aggregate level, through macro-level proxies of intergroup competition, offers little insight into the processes through which ethnic grievances mobilize into contentious action. This article integrates insights from the social movements literature to address how competitive contexts crystallize into broader conflicts. Drawing on data from the civil rights-era Ku (...) Klux Klan—perhaps the quintessential case of contentious ethnic organization in the United States—the analysis focuses on the ways in which meso-level arrangements mediate the relationship between overarching competitive contexts and ethnic conflict. Results of a paired comparative analysis of KKK mobilization in Greensboro and Charlotte demonstrate that social and spatial relations within each city shaped the contours of perceived competition and subsequent ethnic organization in ways that were not always predictable through observation of conventional proxies of competition. (shrink)

Working in ZF+AD alone, we prove that every set of ordinals with cardinality at least Θ can be covered by a set of ordinals in HOD of K (ℝ) of the same cardinality, when there is no inner model with an ℝ-complete measurable cardinal. Here ℝ is the set of reals and Θ is the supremum of the ordinals which are the surjective image of ℝ.

After showing that \ refutes \ for all regular cardinals \, we present a diamond-plus principle \ concerning all subsets of \. Using a forcing argument, we prove that \ holds in Steel’s core model \}}\), an inner model in which the axiom of determinacy can hold. The combinatorial principle \ is then extended, in \}}\), to successor cardinals \ and to certain cardinals \ that are not ineffable. Here \ is the supremum of the ordinals that are the surjective (...) image of the set of reals \. (shrink)

Civil wars vary greatly in their duration. This book argues that conflicts are longer when they involve more actors who can block agreement and identifies specific problems that arise in multi-party bargaining. Quantitative analysis of over 200 civil wars since World War II reveals that conflicts with more of these actors last much longer than those with fewer. Detailed comparison of negotiations in Rwanda and Burundi demonstrates that multi-party negotiations present additional barriers to peace not found in two party conflicts. (...) In addition, conflicts with more veto players produce more casualties, are more likely to involve genocide and are followed by shorter periods of peace. Because they present many barriers to peace, the international community has a poor track record of resolving multi-party conflicts. David Cunningham shows that resolution is possible in these wars if peace processes are designed to address the barriers that emerge in multi-party conflicts. (shrink)

Narrating the Christian life -- Practicing the Christian life -- Living the Christian life.

Mathematical Logic: An Introduction is a textbook that uses mathematical tools to investigate mathematics itself. In particular, the concepts of proof and truth are examined. The book presents the fundamental topics in mathematical logic and presents clear and complete proofs throughout the text. Such proofs are used to develop the language of propositional logic and the language of first-order logic, including the notion of a formal deduction. The text also covers Tarski’s definition of truth and the computability concept. It also (...) provides coherent proofs of Godel’s completeness and incompleteness theorems. Moreover, the text was written with the student in mind and thus, it provides an accessible introduction to mathematical logic. In particular, the text explicitly shows the reader how to prove the basic theorems and presents detailed proofs throughout the book. Most undergraduate books on mathematical logic are written for a reader who is well-versed in logical notation and mathematical proof. This textbook is written to attract a wider audience, including students who are not yet experts in the art of mathematical proof. (shrink)

Given that \(L(\mathbb {R})\models {\text {ZF}}+ {\text {AD}}+{\text {DC}}\), we present conditions under which one can generically add new elements to \(L(\mathbb {R})\) and obtain a model of \({\text {ZF}}+ {\text {AD}}+{\text {DC}}\). This work is motivated by the desire to identify the smallest cardinal \(\kappa \) in \(L(\mathbb {R})\) for which one can generically add a new subset \(g\subseteq \kappa \) to \(L(\mathbb {R})\) such that \(L(\mathbb {R})(g)\models {\text {ZF}}+ {\text {AD}}+{\text {DC}}\).