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A pervasive thought in contemporary philosophy of mathematics is that in order to justify reflection principles, one must hold universism: the view that there is a single universe of pure sets. I challenge this kind of reasoning by contrasting universism with a Zermelian form of multiversism. I argue that if extant justifications of reflection principles using notions of richness are acceptable for the universist, then the Zermelian can use similar justifications. However, I note that for some forms of richness argument, the status of reflection principles as axioms is left open for the Zermelian. [ABSTRACT FROM AUTHOR]

We show how to force, with finite conditions, the forcing axiom PFA(T), a relativization of PFA to proper forcing notions preserving a given Suslin tree T. The proof uses a Neeman style iteration with generalized side conditions consisting of models of two types, and a preservation theorem for such iterations. The consistency of this axiom was previously known using a standard countable support iteration and a preservation theorem due to Miyamoto. [ABSTRACT FROM PUBLISHER]

In this paper, we first introduce the concepts of several separation axioms between the T0 and T1 spaces. We also characterize the modules such as the pm-module, m-module, ε-module, ES-module, D-module and Y-module. Some new results of the prime spectrum of a module M over a commutative ring R are given. [ABSTRACT FROM AUTHOR]

We study model companions of theories extending the graph axioms. First we prove general results concerning the existence of the model companion. Then, by applying these results to the case of graphs, we give a series of companionable and non-companionable examples. [ABSTRACT FROM AUTHOR]

In this paper, using the concept of fuzzy pre-open(fuzzy pre-closed) sets to introduce some new types of fuzzy pre-separation axioms (FPRi space) (for i = 0, 1)stronger forms of fuzzy pre-regular (fuzzy pre-normal) space are introduced and the relationship between these separation axioms and other are investigated. [ABSTRACT FROM AUTHOR]

This paper addresses the issue of stipulation in three cases of implicit definitions (postulates of scientific terms, systems of axioms and abstraction principles). It argues that the alleged implicit definitions do not have a purely stipulative status. Stipulation of the vehicles of the implicit definitions in question should end up with true postulates. However, those postulates should not be taken to be true only in virtue of stipulation since they have extra commitments. Horwich's worry emerges in all three kinds of implicit definitions under consideration, since the existence of meanings so that the alleged postulates are true depends on extra requirements that should be fulfilled. Moreover, if Ramseyfication method is applied to the three kinds of implicit definition, they are split up into two components from which the first one is broadly factual while the second one is purely stipulative. The paper argues that their definitional task in each case should be assigned to their second component i.e. their Carnap-conditional. [ABSTRACT FROM AUTHOR]

Abstract: A mean proximity rule is a voting rule having a mean proximity representation in Euclidean space. Legal ballots are represented as vectors that form the representing polytope. An output plot function determines a location for each possible election output in the same space, and these locations decompose the polytope into proximity regions according to which output is closest. The election outcome is then determined by which region(s) contain the mean position of all ballots cast. Mean neat rules are obtained by relaxing the requirement that the regions be determined by proximity, insisting only that they be neatly separable by a hyperplane. If each of these hyperplanes contains a dense set of rational points (vectors with all rational components), the mean neat voting rule is said to be rational . The aim of this article is to prove that consistency and connectedness are necessary and sufficient conditions for mean neat rationality of any voting rule that is anonymous . Connectedness can be viewed as a strong form of continuity, with an intuitive content related to the Intermediate Value Theorem (or to a discrete analogue of this theorem). The proof relies on a recent result in convexity theory [D. Cervone, W.S. Zwicker, Convex decompositions, J. Convex Anal. 2008 (in press)] and suggests a conjecture: if we relax connectedness to continuity, the class so characterized is that of the mean neat voting rules. This latter class properly contains all intuitive scoring rules. [Copyright &y& Elsevier]

We shall introduce the notions of Λsp-closed and spg-closed sets. We also investigate properties of these sets and introduce some related new separation axioms. [ABSTRACT FROM AUTHOR]