What is Mereology?

This chapter provides a brief illustration of the centrality of part-whole inquiry throughout the history of philosophy, West and East. It explains two original motivations for the contemporary formal explorations of mereological systems. Husserl’s approach, stemming from Brentano, sought to treat part-whole relations as formal ontology – comprising a set of general structural principles applying to any objects whatsoever. Leśniewski’s approach was motivated by nominalism and the search for an alternative foundation for mathematics not beset by the paradoxes of naïve set theory. Some attention is paid to the different uses of ‘part’ in natural language and to whether mereology should be thought of as providing a single, overarching account. The final section details the logical machinery used throughout the book.

Foundations of Transmathematics

Transmathematics has the ambition to be a total mathematics. Many areas of the usual mathematics have been totalised in the transmathematics programme but the totalisations have all been carried out with the usual set theory, ZFC, Zermelo-Fraenkel set theory with the Axiom of Choice. This set theory is adequate but it is, itself, partial. Here we introduce a total set theory as a foundation for transmathematics. Surprisingly we adopt naive set theory. It is usually considered that the Russell Paradox demonstrates that naive set theory is incoherent because an apparently well-specified set, the Russell Set, cannot exist. We dissolve this paradox by showing that the specification of the Russell Set admits many unproblematical sets that do not contain themselves and, furthermore, unequivocally requires that the Russell Set does not contain itself because, were it to do so, that one element of the Russell Set would have contradictory membership. Having resolved the Russell Paradox, we go on to make the case that naive set theory is a paraconsistent logic. In order to demonstrate the sufficiency of naive set theory, as a basis for transmathematics, we introduce the transordinals. The von Neumann ordinals supply the usual ordinals, the simplest unordered set is identical to transreal nullity, and the Russell Set, excluding nullity, is the greatest ordinal, identical to transreal infinity. The generalisation of the transordinals to the whole of established transmathematics is already known. As naive set theory contains all other set theories, it provides a backwardly compatible foundation for the whole of mathematics.

An Essay on Denotational Mathematics

Denotational mathematics is a new rigorous discipline of theoretical computer science that springs out from the attempt to provide a suitable mathematical framework in which laid out new algebraic structures formalizing certain formal patterns coming from computational and natural intelligence, software science, cognitive informatics, neuronal networks, and artificial intelligence. In this chapter, a very brief but rigorous exposition of the main formal structures of denotational mathematics is outlined within naive set theory.

Cantor, Georg (1845–1918)

Georg Cantor and set theory belong forever together. Although Dedekind had already introduced the concept of a set and naïve set theory in 1872, it was Cantor who single-handedly created transfinite set theory as a new branch of mathematics. In a series of papers written between 1874 and 1885, he developed the fundamental concepts of abstract set theory and proved the most important of its theorems. Although today set theory is accepted by the majority of scientists as an autonomous branch of mathematics, and perhaps the most fundamental, this was not always the case. Indeed, when Cantor set out to develop his conception of sets and to argue for its acceptance, he initiated an inquiry into the infinite which raised questions that have still not been completely resolved today.

Bi-Modal Naive Set Theory

This paper describes a modal conception of sets, according to which sets are 'potential' with respect to their members. A modal theory is developed, which invokes a naive comprehension axiom schema, modified by adding `forward looking' and `backward looking' modal operators. We show that this `bi-modal' naive set theory can prove modalized interpretations of several ZFC axioms, including the axiom of infinity. We also show that the theory is consistent by providing an S5 Kripke model. The paper concludes with some discussion of the nature of the modalities involved, drawing comparisons with noneism, the view that there are some non-existent objects.

An Essay on Denotational Mathematics

Denotational mathematics is a new rigorous discipline of theoretical computer science which springs out from the attempt to provide a suitable mathematical framework in which laid out new algebraic structures formalizing certain formal patterns coming from computational and natural intelligence, software science, cognitive informatics, neuronal networks, artificial intelligence. In this chapter, a very brief but rigorous exposition of the main formal structures of denotational mathematics, is outlined within naive set theory.

CAN MODALITIES SAVE NAIVE SET THEORY?

To the memory of Prof. Grigori Mints, Stanford UniversityBorn: June 7, 1939, St. Petersburg, RussiaDied: May 29, 2014, Palo Alto, California