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Ruadhan O'Flanagan

Publications and source records attributed to Ruadhan O'Flanagan.

4 recordsLinked to original sources
arXiv2019

The Constituents of Sets, Numbers, and Other Mathematical Objects, Part One

The sets used to construct other mathematical objects are pure sets, which means that all of their elements are sets, which are themselves pure. One set may therefore be within another, not as an element, but as an element of an element, or even deeper, inside several layers of sets within sets. The introduction of the term constituent to describe a set which is within a given set, however deep, induces an apparently novel partial order on sets, and assigns to any given set a diagram which specifies a directed graph, or category, herein dubbed its constituent structure, indicating which sets within it are constituents of which others. Sets with different numbers of elements can have exactly the same constituent structure. Consequently, constituent structure isomorphisms between sets need not preserve the number of elements, although they are still injective, surjective, and invertible. We consider in detail an example of an isomorphism between a one-element set and a five-element set, which is a surjective mapping despite the mismatch in cardinalities. The constituent structure of a set determines the mathematical objects for which the set is a suitable representation. Different schemes for constructing the natural numbers, such as those of von Neumann and Zermelo, generate sets with the same constituent structures. Objects share the constituent structures, not the elements, of the sets used to construct or represent them. The requirement that an object's properties be faithfully encoded within a set's constituent structure and not its non-constituent characteristics such as its cardinality, when made explicit, dictates a specific and novel way of representing ordered pairs and tuples of sets as sets, providing simple formulae for addressing and extracting sets located deep within nested tuples.

math.LO↗
arXiv2019

The Constituents of Sets, Numbers, and Other Mathematical Objects, Part Two

The arithmetic of natural numbers has a natural and simple encoding within sets, and the simplest set whose structure is not that of any natural number extends this set-theoretic representation to positive and negative integers. The operation that implements addition when applied to sets that represent natural numbers yields both addition and subtraction when used with the sets that encode integers. The encoding of the integers naturally extends beyond them and identifies sets that encode arithmetic expressions and rational numbers. The sets that encode arithmetic expressions naturally specify the set operations that should be performed to evaluate those expressions. The natural encoding of rational numbers within sets expresses each rational number as a novel order-preserving form of continued fraction, which provides a new efficient algorithm for finding rational approximations to irrational numbers. It also arranges all rational numbers within a tree that shows which numbers are constituents of which others. The part of this tree containing the positive rationals coincides with the well-known Stern-Brocot tree, which it extends to all rationals, including zero and negative numbers, introducing new non-trivial symmetries. These symmetries overlap with each other and form a group isomorphic to the group of symmetries of an equilateral triangle.

math.LO↗
arXiv2008

Judgment

The concept of a judgment as a logical action which introduces new information into a deductive system is examined. This leads to a way of mathematically representing implication which is distinct from the familiar material implication, according to which "If A then B" is considered to be equivalent to "B or not-A". This leads, in turn, to a resolution of the paradox of the raven.

math.PR↗