Explore the realm of natural numbers in this 25-minute mathematics lecture. Delve into an unfamiliar aspect of arithmetic with natural numbers, motivated by operations with multisets. Examine natural numbers, including zero, under three operations: addition, union, and intersection. Discover the properties these operations satisfy, including some surprising new distributive laws that form the basis for 'tropical calculus'. Learn about multisets, identity laws, order laws, subtraction, and proofs of various properties. Gain insights into the independence of different laws and explore a variant of the distributive law. Enhance your understanding of fundamental mathematical concepts and their applications in advanced mathematical theories.