Let's give it a try... In the space of computable functions, there is a class X that we would recognize as "having goal G". There is a process SI we would identify as self-improvement. Then converge implies that for nearly any initial function f, the process SI will result in f being in X.

If you want to phrase this in an updateless way, say that "any function with property SI is in X", defining X as "ultimately having goal G".