Explore the concept of exponentiation, or "caret," in the context of multiset arithmetic in this 35-minute mathematics lecture. Learn about the extension of arithmetic operations from addition and multiplication to a higher-level operation. Discover a new approach to arithmetic using multisets, building on previous foundational concepts. Examine the inductive sequence of "counting functions" corresponding to the hierarchy of Zero, Nat, Poly, and Multi. Gain insights into this novel arithmetic system through numerous concrete examples and illustrations. Understand the recursive nature of the approach and its potential for further extension to higher levels of operations.