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Extraordinary Ordinals

Syntax. E x p r e s s i o n s e , b , f , a ∷ = x V a r i a b l e ∣ x ⇒ b A b s t r a c t i o n ∣ f ← a A p p l i c a t i o n V a r i a b l e s x ∈ k , x , y , … N u m b e r s n ∈ 0 , 1 , 2 , … \small\begin{array}{lrcll} \mathrm{Expressions}\quad & e,b,f,a & {\Coloneqq} & x & \quad\mathrm{Variable} \\ & & {\mid} & x\Rightarrow b & \quad\mathrm{Abstraction} \\ & & {\mid} & f\leftarrow a & \quad\mathrm{Application} \\ \mathrm{Variables} & x & {\in} & \mathsf{k}, \mathsf{x}, \mathsf{y}, \dots \\ \mathrm{Numbers} & n & {\in} & 0, 1, 2, \dots \end{array} Expressions Variables Numbers e , b , f , a x n : : = ∣ ∣ ∈ ∈ x x ⇒ b f ← a k , x , y , … 0 , 1 , 2 , … Variable Abstraction Application There are 3 categories— Linear, Affine, and Non-Linear—each having several encodings.

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