The Fraenkel-Mostowski method has been widely used to prove independence results among weak versions of the axiom of choice. In this paper it is shown that certain statements cannot be proved by this method. More specifically it is shown that in all Fraenkel-Mostowski models the following hold: 1. The axiom of choice for sets of finite sets implies the axiom of choice for sets of well-orderable sets. 2. The Boolean prime ideal theorem implies a weakened form of Sikorski's theorem.
Let NBG be von Neumann-Bernays-Gödel set theory without the axiom of choice and let NBGA be the modification which allows atoms. In this paper we consider some of the well-known class or global forms of the wellordering theorem, the axiom of choice, and maximal principles which are known to be equivalent in NBG and show they are not equivalent in NBGA.
This is a continuation of . We study the Tychonoff Compactness Theorem for various definitions of compactness and for various types of spaces . We also study well ordered Tychonoff products and the effect that the multiple choice axiom has on such products.
We study statements about countable and well-ordered unions and their relation to each other and to countable and well-ordered forms of the axiom of choice. Using WO as an abbreviation for “well-orderable”, here are two typical results: The assertion that every WO family of countable sets has a WO union does not imply that every countable family of WO sets has a WO union; the axiom of choice for WO families of WO sets does not imply that the countable union (...) of countable sets is WO. (shrink)
We investigate the set theoretical strength of some properties of normality, including Urysohn's Lemma, Tietze-Urysohn Extension Theorem, normality of disjoint unions of normal spaces, and normality of Fσ subsets of normal spaces.
We show that it is not possible to construct a Fraenkel-Mostowski model in which the axiom of choice for well-ordered families of sets and the axiom of choice for sets are both true, but the axiom of choice is false.
Two Fraenkel-Mostowski models are constructed in which the Boolean Prime Ideal Theorem is true. In both models, AC for countable sets is true, but AC for sets of cardinality 2math image and the 2m = m principle are both false. The Principle of Dependent Choices is true in the first model, but false in the second.
In Zermelo-Fraenkel set theory with the axiom of choice every set has the same cardinal number as some ordinal. Von Rimscha has weakened this condition to “Every set has the same cardinal number as some transitive set”. In set theory without the axiom of choice, we study the deductive strength of this and similar statements introduced by von Rimscha.
The apolipoprotein E gene plays an important role in the pathogenesis of Alzheimer's disease , and amyloid plaque comprised mostly of the amyloid-beta peptide ) is one of the major hallmarks of AD. However, the relationship between these two important molecules is poorly understood. We examined how A treatment affects APOE expression in cultured cells and tested the role of the transcription factor NF-B in APOE gene regulation. To delineate NF-B's role, we have characterized a 1098 nucleotide segment containing the (...) 5'-flanking region of the human APOE gene . Sequence analysis of this region suggests the presence of two potential NF-B elements. To demonstrate promoter activity, the region was cloned upstream of a promoterless luciferase gene. This segment was able to drive expression of luciferase in transient transfections of human fetal glial cells. Promoter activity was stimulated twofold by A treatment. Pretreatment with double-stranded DNA decoy oligonucleotides against NF-B reduced A stimulation. Deletion and mutagenetic analyses demonstrated that the distal NF-B element was functional and showed a strong DNA-protein complex band in gel shift analysis, similar to that from control NF-B consensus element. An anti-inflammatory and anti-NF-B drug, sodium salicylate, significantly blocked A-induced APOE promoter function. Our data provide evidence that upregulation of APOE by A in astroglial cells is mediated by an NF-B-element present in the 5'-flanking region of the APOE gene. (shrink)
One philosophical approach to causation sees counterfactual dependence as the key to the explanation of causal facts: for example, events c (the cause) and e (the effect) both occur, but had c not occurred, e would not have occurred either. The counterfactual analysis of causation became a focus of philosophical debate after the 1973 publication of the late David Lewis's groundbreaking paper, "Causation," which argues against the previously accepted "regularity" analysis and in favor of what he called the "promising alternative" (...) of the counterfactual analysis. Thirty years after Lewis's paper, this book brings together some of the most important recent work connecting--or, in some cases, disputing the connection between--counterfactuals and causation, including the complete version of Lewis's Whitehead lectures, "Causation as Influence," a major reworking of his original paper. Also included is a more recent essay by Lewis, "Void and Object," on causation by omission. Several of the essays first appeared in a special issue of the Journal of Philosophy, but most, including the unabridged version of "Causation as Influence," are published for the first time or in updated forms.Other topics considered include the "trumping" of one event over another in determining causation; de facto dependence; challenges to the transitivity of causation; the possibility that entities other than events are the fundamental causal relata; the distinction between dependence and production in accounts of causation; the distinction between causation and causal explanation; the context-dependence of causation; probabilistic analyses of causation; and a singularist theory of causation. (shrink)
Written over the last 18 months of his life and inspired by his interest in G. E. Moore's defence of common sense, this much discussed volume collects Wittgenstein's reflections on knowledge and certainty, on what it is to know a proposition for sure.