A Multiset Approach to Arithmetic - Math Foundations

Explore a new framework for basic arithmetic and algebra using multisets in this 39-minute lecture. Learn how to define natural numbers, polynumbers, and multinumbers using the concept of multisets, which are unordered collections allowing repetition. Discover inductive definitions for these number systems and understand how arithmetic operations can be generalized within this framework. Examine the closure properties of different number types under addition and multiplication, and investigate the relationship between multisets and computer science. Gain insights into the historical development of this approach and its potential implications for mathematical foundations.

Introduction and history of multiset development
A multiset mset is an unordered collection allowing repetitions
A natural number NAT is an mset of zeroes
A polynumber is an mset of natural numbers
A multinumber is an mset of polynumbers
Addition of msets
NAT is closed under addition and commutative, associative
Multinumbers are also closed under addition
Multiplication of msets of msets
Each "type domain" is closed under addition and multiplication
The meaning of "poly"
Distinction of mset and list
Mathematics as a topic in computer science