Extraordinary Ordinals

Expressionse,b,f,a∷=xVariable∣x⇒bAbstraction∣f←aApplicationVariablesx∈k,x,y,…Numbersn∈0,1,2,… \small\begin{array}{lrcll} \mathrm{Expressions}\quad &amp;amp; e,b,f,a &amp;amp; {\Coloneqq} &amp;amp; x &amp;amp; \quad\mathrm{Variable} \\ &amp;amp; &amp;amp; {\mid} &amp;amp; x\Rightarrow b &amp;amp; \quad\mathrm{Abstraction} \\ &amp;amp; &amp;amp; {\mid} &amp;amp; f\leftarrow a &amp;amp; \quad\mathrm{Application} \\ \mathrm{Variables} &amp;amp; x &amp;amp; {\in} &amp;amp; \mathsf{k}, \mathsf{x}, \mathsf{y}, \dots \\ \mathrm{Numbers} &amp;amp; n &amp;amp; {\in} &amp;amp; 0, 1, 2, \dots \end{array} ExpressionsVariablesNumbers​e,b,f,axn​::=∣∣∈∈​xx⇒bf←ak,x,y,…0,1,2,…​VariableAbstractionApplication​