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

Abstract: 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.