The Refutation of Singularism?

Traditional analyses of plurals tended to eliminate plural expressions in favor of singular ones. These “singularist” analyses have recently faced many objections, which are intended to provide indirect support for the alternative analysis provided by plural logic. This chapter evaluates four such objections and concludes that they are less compelling than is often assumed. This conclusion is borne out by a close examination of various plural versions of Cantor’s theorem.

Cantor’s theorem

Cantor’s theorem states that the cardinal number (‘size’) of the set of subsets of any set is greater than the cardinal number of the set itself. So once the existence of one infinite set has been proved, sets of ever increasing infinite cardinality can be generated. The philosophical interest of this result lies (1) in the foundational role it played in Cantor’s work, prior to the axiomatization of set theory, (2) in the similarity between its proof and arguments which lead to the set-theoretic paradoxes, and (3) in controversy between intuitionist and classical mathematicians concerning what exactly its proof proves.

Cousin’s Lemma

Summary We formalize, in two different ways, that “the n-dimensional Euclidean metric space is a complete metric space” (version 1. with the results obtained in [13], [26], [25] and version 2., the results obtained in [13], [14], (registrations) [24]). With the Cantor’s theorem - in complete metric space (proof by Karol Pąk in [22]), we formalize “The Nested Intervals Theorem in 1-dimensional Euclidean metric space”. Pierre Cousin’s proof in 1892 [18] the lemma, published in 1895 [9] states that: “Soit, sur le plan YOX, une aire connexe S limitée par un contour fermé simple ou complexe; on suppose qu’à chaque point de S ou de son périmètre correspond un cercle, de rayon non nul, ayant ce point pour centre : il est alors toujours possible de subdiviser S en régions, en nombre fini et assez petites pour que chacune d’elles soit complétement intérieure au cercle correspondant à un point convenablement choisi dans S ou sur son périmètre.” (In the plane YOX let S be a connected area bounded by a closed contour, simple or complex; one supposes that at each point of S or its perimeter there is a circle, of non-zero radius, having this point as its centre; it is then always possible to subdivide S into regions, finite in number and sufficiently small for each one of them to be entirely inside a circle corresponding to a suitably chosen point in S or on its perimeter) [23]. Cousin’s Lemma, used in Henstock and Kurzweil integral [29] (generalized Riemann integral), state that: “for any gauge δ, there exists at least one δ-fine tagged partition”. In the last section, we formalize this theorem. We use the suggestions given to the Cousin’s Theorem p.11 in [5] and with notations: [4], [29], [19], [28] and [12].

TRANSFINITE CARDINALS IN PARACONSISTENT SET THEORY

This paper develops a (nontrivial) theory of cardinal numbers from a naive set comprehension principle, in a suitable paraconsistent logic. To underwrite cardinal arithmetic, the axiom of choice is proved. A new proof of Cantor’s theorem is provided, as well as a method for demonstrating the existence of large cardinals by way of a reflection theorem.