[hyperion2010]

Pretty sure the incompleteness theorem doesn't enable one to study things one cannot study. (eg imagine a logic for which one could not write down the axioms but was nonetheless 'consistent' by some external measure)

[elektromekatron]

Yeah, but you made a statement about the kind of logic you cannot study, so we can begin by formalizing that.

[hyperion2010]

I suppose an easy way out is to show that such a thing cannot exist in the way that other mathematical structures exist, trivially showing that you have studied all of it that there is(n't) to study.