User blog comment:Syst3ms/STON 2 : BMS and the Star/@comment-35470197-20190913011017

> I would also like to address this question from a definition standpoint : what in the notation makes the notation strong? My immediate answer is the lev function.

> intuitively, how many times it can be put inside of a psi subscript before becoming singular.

The lev function would be one of the factors of the strength, but I guess that the existence of such an index indecating "how many times it can be collapsed" is not the main factor if it actually as strong as expected.

Indeed, Rathjen's standard OCF with weakly compact cardinal already has a similar structure (i.e. the index \(\xi\) in \(\Psi^{\xi}_{\pi}(\alpha)\). Arai's OCF has more complicated structures, which I have not fully understood yet.

As Taranovsky's method to use abstract "degree" structure in order to define a new binary relation shows that the strength of an associated collapsing functions heavily depends on a structure to indicate "how many times", the lev function would possess a special property other than indecating "how many times".