Extraordinary Ordinals

The author presents most known numeral systems (ways of representing natural numbers) in lambda calculus, classified by whether the term use their bound variables exactly one time (linear), at most one time (affine), or multiple times (non-linear). He illustrates some numerals in each system with a graphical notation that strongly reminds me of interaction nets [1], a computational model closely related to lambda calculus. The notation they use for lambda terms is rather non-standard. Compare

> In β-reduction, k[(x⇒b)←a]⊳k[b{a/x}]k[(x⇒b)←a]⊳k[b{a/x}]

with Wikipedia's [2]

> The β-reduction rule states that a β-redex, an application of the form (λx. t) s, reduces to the term t[x:=s].

The k[...] part means that β-reduction steps can happen in arbitrary contexts.

[1] https://en.wikipedia.org/wiki/Interaction_nets

[2] https://en.wikipedia.org/wiki/Lambda_calculus

Hmm nice I guess, but I expected it was going to be about transfinite ordinals. I wonder if it can be extended to them.

I didn’t understand that notation. Can someone please explain?

I think I lack context to see what this is about. The line graphs are pretty though, and I'd like to understand more.

This is beautiful art

This should be "numerals"