User blog:ChromaticiT/Infinite-order Xi Function

The ITTM Busy beaver is to the Busy beaver as the Infinite-order Xi function is to the Xi function.

We define a combinatory language with countably infinite symbols such that there is a bijection between these symbols and the natural numbers (including 0).

0x := $$\iota$$x as in Chris Barker's iota combinator. Nxyzw := Where N is a natural number denoting a symbol, $$\Omega$$xyz (as in the oracle combinator) if x is well founded, w otherwise.

Now that we have our language, let's just reuse the definition for the Xi function.

If we start with a string of n symbols and we beta-reduce it, the largest possible finite output is called $$\Xi_{\infty}$$(n)