Where can I find information about metametalogic

Question

I have been looking for information about metametalogic (theory about metalogic) but I found basically nothing. I would be grateful if anybody could refer me to a book or a publication. thanks.

Answer 1

Metalogic, in a common understanding of that somewhat variously used term, is mathematics applied to theories of interest to logicians (so the logician's theories are themselves treated as formal, mathematical objects).

Now most metalogic is done as informal maths. What's the informal toolkit we need to talk about theories-as-mathematical-objects? Stuff about formal strings (equivalent to arithmetic), the theory of computability, stuff about ordinals (for talking about proof theory), stuff about sets more generally (for when we are dealing with model theory), and such like. So a metatheory for informal metalogic will be a theory about such theories of arithmetic, computation, ordinals, sets and stuff now treated as mathematical objects. But those are already the theories of interest to logicians one level down, so to speak, i.e. are already the topic of metalogic.

So you could say that metametalogic just is metalogic again!

Which I guess is why you don't find people talking too much about metametalogic, as if there is a separate discipline waiting to be explored.