A multiset approach to arithmetic | Math Foundations 227 | N J Wildberger

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.

I'd like to thank my Patreon supporters for encouragement and helpful comments.

Video Content: (Thanks to phi Architect)

00:00 Introduction and history of multiset development
4:20 A multiset (mset) is an unordered collection allowing repetitions
7:20 A natural number (NAT) is an mset of zeroes
9:40 A polynumber is an mset of natural numbers
11:55 A multinumber is an mset of polynumbers
14:40 Addition of msets
19:15 NAT is closed under addition and commutative, associative
21:40 Multinumbers are also closed under addition
22:20 Multiplication of msets of msets
31:00 Each "type domain" is closed under addition and multiplication
32:30 The meaning of "poly"
36:15 Distinction of mset and list
38:30 Mathematics as a topic in computer science

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