Insights into Mathematics | The operation of caret / exponentiation (new!) via multisets | Math Foundations 229 | N J Wildberger @njwildberger | Uploaded 1 year ago | Updated 8 hours ago
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 the Insights into Mathematics Playlists:

youtube.com/playlist?list=PL55C7C83781CF4316
youtube.com/playlist?list=PL3C58498718451C47
youtube.com/playlist?list=PL5A714C94D40392AB
youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
youtube.com/playlist?list=PLIljB45xT85D94vHAB8joyFTH4dmVJ_Fw
youtube.com/playlist?list=PL8403C2F0C89B1333
youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
youtube.com/playlist?list=PLIljB45xT85DH__ZzGQWQrVRxlbKh-Nsa
youtube.com/playlist?list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
youtube.com/playlist?list=PL6763F57A61FE6FE8
youtube.com/playlist?list=PLBF39AFBBC3FB30AF
youtube.com/playlist?list=PLIljB45xT85DSrlV6NX8RMBksZhdTHtwW
youtube.com/playlist?list=PLIljB45xT85Bmcc9ksBOAKgIZAl0BwPg7

Here are the Wild Egg Maths Playlists (some available only to Members!)

youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
youtube.com/playlist?list=PLzdiPTrEWyz4VlOppC5CN0D0GjrjvBGKy
youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-
youtube.com/playlist?list=PLzdiPTrEWyz6zpIZ4Y_RK9zyJ9OufqNJv
youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
youtube.com/playlist?list=PLzdiPTrEWyz5j1BJdXBw1MFst_nQAqzZ_
youtube.com/playlist?list=PLzdiPTrEWyz5HWgaVkhIwpGVKi6fciRxW
youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
youtube.com/playlist?list=PLzdiPTrEWyz7LKbuJAHaXAhRj2ylD0OoX
youtube.com/playlist?list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
youtube.com/playlist?list=PLzdiPTrEWyz6MwUTOHRgC0oIxVtaHGQBd
youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
youtube.com/playlist?list=PLzdiPTrEWyz4IknbwXMEVwxOWS3z1Ch3C
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