Hyper-extensions in metric fixed point theory

We apply a modern axiomatic system of nonstandard analysis in metric fixed point theory. In particular, we formulate a nonstandard iteration scheme for nonexpansive mappings and present a nonstandard approach to fixed-point problems in direct sums of Banach spaces.Comment: 12 page

Iterated hyper-extensions and an idempotent ultrafilter proof of Rado’s Theorem

By using nonstandard analysis, and in particular iterated hyperextensions, we give foundations to a peculiar way of manipulating ultrafilters on the natural numbers and their pseudo-sums. The resulting formalism is suitable for applications in Ramsey theory of numbers. To illustrate the use of our technique, we give a (rather) short proof of Milliken-Taylor’s Theorem and a ultrafilter version of Rado’s Theorem about partition regularity of diophantine equations

Nonstandard Methods in Ramsey Theory and Combinatorial Number Theory

The goal of this present manuscript is to introduce the reader to the nonstandard method and to provide an overview of its most prominent applications in Ramsey theory and combinatorial number theory.Comment: 126 pages. Comments welcom

Back to the events themselves: on what events are and how we perceive them

A task confronting all the theoretical branches of philosophy is Peacocke’s Integration Challenge: “providing, for a given area, a simultaneously acceptable metaphysics and epistemology, and showing them to be so” (1999, 1). When it comes to the study of everyday events such as basketball shots and ripples on a pond, there is a phenomenological analogue of the Integration Challenge: the twofold task of explicating both the nature of such events and the way they show up in our perceptual consciousness. In this dissertation, I propose an account of events in response to the twofold challenge. On the one hand, an event is a complex entity constituted from tropes in accordance with its kind. On the other hand, our perceptual experiences of events differ from our other varieties of experiences because they uniquely feature the awareness of certain temporal phenomena that function as the boundaries between events. The dissertation is divided into five chapters. After a short introductory chapter, I develop a metaphysical theory of events. I follow a framework based on Evnine (2016) and divide theories of events into two groups: those that invoke the constitution relation and those that do not. Chapter 2 reviews and argues against the major theories in the first group, whereas Chapter 3 defends my own view against several alternatives in the second group. Roughly, my view is that a number of events jointly constitute another if and only if the event-kind the latter falls under makes the latter dependent upon the former. The remaining chapters defend a Husserlian view about event perception. Chapter 4 situates my favored view in the theoretical landscape. I argue for a representational view by drawing on Husserlian ideas, in particular the idea that any perceptual content has an expectational component in addition to a component that represents what is strictly visible. Chapter 5 expands on the view and supplies an analysis of event perception. By combining the theory of perceptually based expectations proposed by Yoshimi (2016) with results from linguistics and psychology, I argue that event perception can be better understood with the theoretical apparatus of possible worlds

An axiomatic presentation of the nonstandard methods in mathematics

A nonstandard set theory *ZFC is proposed that axiomatizes the nonstandard embedding *. Besides the usual principles of nonstandard analysis, all axioms of ZFC except regularity are assumed. A strong form of saturation is also postulated. *ZFC is a conservative extension of ZFC