User blog comment:Emlightened/New(ish) Concept: Googological Systems/@comment-5529393-20170730195718/@comment-27513631-20170730211953

I'll go check that.

I'm pretty sure that I can deliver. The basic idea is to use inductive types with orderings, and allow self-reference in the definition of the function as long as the function takes a smaller input when referenced by itself. This should guarantee halting if the orders are well founded. The stronger version weakens the embeddability property.

The SVO tree can be simplified by letting the base function for \(\phi(\cdots)\) be \(\phi = (\alpha \mapsto 1 + \alpha)\). Additionally, if we stay less than SVO as opposed to trying to achieve SVO, then we can just consider n-ary trees, as extensions to a notation where binary trees give \(\varepsilon_0\).

I'm not particularly sure how Bar Recursion can be used, however I have noticed parallels between proof that it halts and trying to create efficient data storage for exact numbers of googological magnitude. I believe that some notion of recursing on higher-order functions may be required to elegantly surpass BHO though, as Rank 1 (first order) Bar Recursion has strength BHO.