A Multiset Approach to Arithmetic (Mathematics Foundations 227)
Associate Professor of Mathematics
This lecture reconstructs fundamental arithmetic operations—addition and multiplication—within a multiset theoretic framework, defining natural numbers, polynumbers, and multinumbers as nested multisets. It explores closure properties and distinct algebraic structures.
We introduce a new framework for basic arithmetic and algebra, using the data structure of a multiset, or mset. This is an unordered collection of mathematical objects in which repetition is allowed. But what constitutes a "mathematical object"? One way of proceeding is to begin with the simplest possible mset, which is the empty mset, and build up from there. This is analogous to well -known treatments from set theory, but the consequences when we use msets instead are profound.
We are able to give simple and natural inductive definitions of natural numbers, polynumbers, and higher extensions which we call multinumbers. Crucially the operations are also defined in a very general fashion.