We show that if the weak compactness of a cardinal is made indestructible by means of any preparatory forcing of a certain general type, including any forcing naively resembling the Laver preparation, then the cardinal was originally supercompact. We then apply this theorem to show that the hypothesis of supercompactness is necessary for certain proof schemata.

First Published in 1999. Routledge is an imprint of Taylor & Francis, an informa company.

It is widely agreed that a new conception of history was developed in the early nineteenth century: the past came to be seen in a new light, as did the way of studying the past. This article discusses the nature of this collective revision, focusing on one of its first and most important manifestations: Ranke's 1824 Geschichten der romanischen und germanischen Völker. It argues that, in Ranke's case, the driving force of the revision was religious, and that, subsequently, an understanding (...) of the nature of Ranke's religious attitude is vital to any interpretation of his historical revision. Being aesthetic-experiential rather than conceptual or "positive," this religious element is reflected throughout Ranke's enterprise, in source criticism and in historical representation no less than in the conception of cause and effect in the historical process. These three levels or aspects of the historical enterprise correspond to the experience of the past, and are connected by the essence of the experience: visual perception. The highly individual character of the enterprise, its foundation in sentiments and experiences of little persuasive force that only with difficulty can be brought into language at all, explains the paradoxical nature of the Rankean heritage. On the one hand, Ranke had a great and lasting impact; on the other hand, his approach was never re-utilized as a whole, only in its constituent parts—which, when not in the relationship Ranke had envisioned, took on a new and different character. This also suggests the difference between Ranke's revision and a new paradigm: whereas the latter is an exemplary solution providing binding regulations, the former is unrepeatable. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The humble origins of Russell's paradox by J. Alberto Coffa ON SEVERAL OCCASIONS Russell pointed out that the discovery of his celebrated paradox concerning the class of all classes not belonging to themselves was intimately related to Cantor's proof that there is no greatest cardinal. lOne of the earliest remarks to that effect occurs in The Principles ofMathematics where, referring to the universal class, the class of all classes (...) and the class of all propositions, he notes that when we apply the reasoning of his [Cantor's] proofto the cases in question we find ourselves met by definite contradictions, of which the one discussed in Chapter x is an example. (P. 362) And in a footnote he adds: "It was in this way that I discovered this contradiction". Throughout his writings Russell left a number ofhints concerning the sort of connection he had drawn between Cantor's proof and his own discovery. In fact, his suggestions are so specific that there would seem to be little room left for speculation concerning how the discovery took place.2 The picture that emerges almost immediately from Russell's observations is the following. For reasons which are 1 See, e.g., Bertrand Russell, The Principles of Mathematics, 2nd ed. (London: Allen & Unwin, 1937), §§100, 344-9; G. Frege, Wissenschaftlicher Briefwechsel (Hamburg: Felix Meiner Verlag, 1976), pp. 215-16; B. Russell, Essays in Analysis, ed. D. Lackey (New York: George Braziller, 1973), p. 139; B. Russell, Introducrion to Mathemarical Philosophy (New York: Simon and Schuster, 1971), p. 136; B. Russell, My Philosophical Development (New York: Simon and Schuster, 1959), pp. 75-6; The Autobiography of Bertrand Russell, I (New York: Bantam Books, 1968), 195. 2 See, e.g., Ch. Thiel's "Einleitung des Herausgebers" in Frege, Wissenschaftlicher Briefwechsel, pp. 203 and 216 (footnote), and 1. Grattan-Guinness, "How Bertrand Russell Discovered his Paradox", Historia Marhematica, 5 (1978),127-37. 31 32 Russell, nos. 33-4 (Spring-Summer 1979) never made clear, Russell decided to analyze Cantor's argument applying it to "large" classes such as the universal class V, and the class of all classes. When, for example, we consider V, its power set and the correlationf(x) = {x} if x is not a class, f(x) =. x otherwise, then Cantor's diagonal class D turns out to be the class of all classes not belonging to themselves. Moreover, since the element ofV which is taken toD byfisD itself(i.e., sincef(D) = D), Cantor's reasoning invites us to raise the question whether D belongs tof(D) (i.e., toD) or not; and it establishes that it does precisely if it doesn't.3 The purpose of this note is to present recently uncovered information that complements and corrects our present understanding of Russell's discovery of his paradox. As it turns out, far from originating from his desire to apply the ideas in Cantor's theorem, a version of Russell's paradox first occurred in an argument that Russell had devised, late in 1900, in order to establish the invalidity of that theorem. An appeal to "large" classes such as V or Class was, as we shall see, essential to Russell's attempted refutation. As he displayed the details of his counterexample Russell's paradox emerged, at first unrecognized, as the by-product of a project that the paradox itself would eventually undermine. In a paper written in 1901 and largely devoted to a popular exposition and defence of Cantor's theory of the infinite, Russell expressed what appeared to be a rpinor reservation to Cantor's treatment : There is a greatest of all infinite numbers, which is the number of things altogether, of every sort and kind. It is obvious that there cannot be a greater number than this, because, if everything has been taken, there is nothing left to add. Cantor has a proof that there is no greatest number, and if this proof were valid, the contradictions of infinity would reappear in a sublimated form. But in this one point, the master has been guilty of a very subtle fallacy, which I hope to explain in some future work.4 3 See Principles, §349. Perhaps I should remind... (shrink)

The focus of this study was to analyze the content, distribution, and comparative genome relationships of 996 chromosome bin-mapped expressed sequence tags accounting for 2266 restriction fragments on the homoeologous group 3 chromosomes of hexaploid wheat. Of these loci, 634, 884, and 748 were mapped on chromosomes 3A, 3B, and 3D, respectively. The individual chromosome bin maps revealed bins with a high density of mapped ESTs in the distal region and bins of low density in the proximal region of the (...) chromosome arms, with the exception of 3DS and 3DL. These distributions were more localized on the higher-resolution group 3 consensus map with intermediate regions of high-mapped-EST density on both chromosome arms. Gene ontology classification of mapped ESTs was not significantly different for homoeologous group 3 chromosomes compared to the other groups. A combined analysis of the individual bin maps using 537 of the mapped ESTs revealed rearrangements between the group 3 chromosomes. Approximately 232 of the consensus mapped ESTs matched sequences on rice chromosome 1 and revealed large- and small-scale differences in gene order. Of the group 3 mapped EST unigenes ∼21 and 32% matched the Arabidopsis coding regions and proteins, respectively, but no chromosome-level gene order conservation was detected. (shrink)