The well-ordering of sets

Thesis (M.A.)--Boston Universit

Ernst Schroeder and Zermelo’s Anticipation of Russell’s Paradox

Ernst Zermelo presented an argument showing that there is no set of all sets that are members of themselves in a letter to Edmund Husserl on April 16th of 1902, and so just barely anticipated the same contradiction in Betrand Russell’s letter to Frege from June 16th of that year. This paper traces the origins of Zermelo’s paradox in Husserl’s criticisms of a peculiar argument in Ernst Schroeder’s 1890 Algebra der Logik. Frege had also criticized that argument in his 1985 “A Critical Elucidation of Some Points in E. Schroeder Vorlesungen über die Algebra der Logik”, but did not see the paradox that Zermelo found. Alonzo Church, in “Schroeder’s Anticipation of the Simple Theory of Types” from 1939, cricized Frege’s treatment of Schroeder’s views, but did not identify the connection with Russell’s paradox

Ordering infinite utility streams comes at the cost of a non-Ramsey set.

The existence of a Paretian and finitely anonymous ordering in the set of infnite utility streams implies the existence of a non-Ramsey set (a nonconstructive object whose existence requires the axiom of choice). Therefore, each Paretian and finitely anonymous quasi-ordering either is incomplete or does not have an explicit description. Hence, the possibility results of Svensson (1980) and of Bossert, Sprumont, and Suzumura (2006) do require the axiom of choice.Intergenerational justice; Pareto; Multi-period social choice; Axiom of choice; Constructivism;

Ordering infinite utility streams comes at the cost of a non-Ramsey set

The existence of a Paretian and finitely anonymous ordering in the set of infinite utility streams implies the existence of a non-Ramsey set (a nonconstructive object whose existence requires the axiom of choice). Therefore, each Paretian and finitely anonymous quasi-ordering either is incomplete or does not have an explicit description. Hence, the possibility results of Svensson (1980) and of Bossert, Sprumont, and Suzumura (2006) do require the axiom of choice.Intergenerational justice; Pareto; Multi-period social choice; Axiom of choice; Constructivism.

Caristi's Fixed Point Theorem and Selections of Set-Valued Contractions

AbstractWe show (without the axiom of choice) that the Zermelo theorem implies directly a restriction of the Caristi fixed point theorem to continuous functions. Under the axiom of choice, this restriction is proved to be equivalent to Caristi's theorem. We also discuss Kirk's problem on an extension of the Caristi theorem and we establish two selection theorems for set-valued contractions

From axiomatization to generalizatrion of set theory

The thesis examines the philosophical and foundational significance of Cohen's Independence results. A distinction is made between the mathematical and logical analyses of the "set" concept. It is argued that topos theory is the natural generalization of the mathematical theory of sets and is the appropriate foundational response to the problems raised by Cohen's results. The thesis is divided into three parts. The first is a discussion of the relationship between "informal" mathematical theories and their formal axiomatic realizations this relationship being singularly problematic in the case of set theory. The second part deals with the development of the set concept within the mathemtical approach. In particular Skolem's reformulation of Zermlelo's notion of "definite properties". In the third part an account is given of the emergence and development of topos theory. Then the considerations of the first two parts are applied to demonstrate that the shift to topos theory, specifically in its guise of LST (local set theory), is the appropriate next step in the evolution of the concept of set, within the mathematical approach, in the light of the significance of Cohen's Independence results

Poincaré's philosophy of mathematics

The primary concern of this thesis is to investigate the explicit philosophy of mathematics in the work of Henri Poincare. In particular, I argue that there is a well-founded doctrine which grounds both Poincare's negative thesis, which is based on constructivist sentiments, and his positive thesis, via which he retains a classical conception of the mathematical continuum. The doctrine which does so is one which is founded on the Kantian theory of synthetic a priori intuition. I begin, therefore, by outlining Kant's theory of the synthetic a priori, especially as it applies to mathematics. Then, in the main body of the thesis, I explain how the various central aspects of Poincare's philosophy of mathematics - e.g. his theory of induction; his theory of the continuum; his views on impredicativiti his theory of meaning - must, in general, be seen as an adaptation of Kant's position. My conclusion is that not only is there a well-founded philosophical core to Poincare's philosophy, but also that such a core provides a viable alternative in contemporary debates in the philosophy of mathematics. That is, Poincare's theory, which is secured by his doctrine of a priori intuitions, and which describes a position in between the two extremes of an "anti-realist" strict constructivism and a "realist" axiomatic set theory, may indeed be true

Did Gödel prove that we are not machines? (On philosophical consequences of Gödel's theorem)

Gödel's incompleteness theorem has been the most famous example of a mathematical theorem from which deep philosophical consequences follow. They are said to give an insight, first, into the nature of mathematics, and more generally of human knowledge, and second, into the nature of the mind. The limitations of logicist or formalist programmes of mathematics have had a clear significance against the background of the foundational schools of the early decades of this century. The limitations of mechanism, or of the vision underlying research in the field of Artificial Inteligence, gain significance only now. Yet, while the limitations imposed by Gödel's theorem upon the extent of formal methods seem unquestionable they seem to have very little to say about the restrictions concerning mathematical or computer practice. And the alleged consequences concerning the non-mechanical character of human mind are questionable. The standard reasoning, known as Lucas' argument, begs the question, and actually implies that Lucas is inconsistent