The operation of caret / exponentiation (new!) via multisets | Math Foundations 229 | N J Wildberger

We want to extend our ladder of operations: addition, then multiplication then to a new higher operation which we will call "exponentiation" or better yet following the suggestion of viewer Cristopher Game simply “caret” in accord with the wedge or caret symbol.

This is in the context of a somewhat new arithmetic with multisets that we laid out in the two previous videos (MF 227 and MF 228), and that does not suppose any familiarity with prior arithmetical systems.

Along the way in this video we introduce an inductive sequence of "counting functions" that correspond to the hierarchy of Zero to Nat to Poly to Multi that we have established so far. The inductive / recursive aspect of this approach means that it is pretty easy to see how to extend what we are doing to further higher "levels".

We try to illustrate this new way of thinking about arithmetic in a concrete way with lots of examples.

PLEASE NOTE: in subsequent lectures I will shift to the use of the "caret" terminology: that means that what we are calling "the exponential of A and B" in this lecture will be replaced with "the caret of A and B". So we will use caret also as a verb: "to caret multisets", in the same spirit as "to add multisets" or "to multiply multisets". We will not use carrots however.

Happy New Year 2023!

Video Content: (thanks to phi Architect)
0:00 Introduction - historical context of sets
2:20 Recursive structures in msets
4:30 Review of types
7:00 Elements of an mset
7:45 General mset notation
8:50 Examples of msets and operations
10:45 Counting functions on pure msets
11:30 Def (Z)
11:45 Def (N)
13:40 Def (P)
14:50 Def (M)
19:15 Operations on msets - addition, multiplication
20:45 Caret operation - exponentiation
26:30 A slightly more complicated example
29:40 Properties of the caret operation
34:30 It's all trees

***********************
Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: https://www.youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: https://www.youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: https://www.youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: https://www.youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: https://www.youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: https://www.youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************

And here are some of the Wild Egg Maths Playlists (some available only to Members!)

Solving Polynomial Equations: https://www.youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
Exceptional Structures in Maths and Physics via Dynamics on Graphs" https://www.youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
Triangle Centres: https://www.youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: https://www.youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-

Видео The operation of caret / exponentiation (new!) via multisets | Math Foundations 229 | N J Wildberger канала Insights into Mathematics