Does consciousness itself suffer from intractability similar to Gödel's incompleteness theorems? Is there a consistent set of axioms that can even include all t

Does consciousness itself suffer from intractability similar to Gödel's incompleteness theorems? Is there a consistent set of axioms that can even include all the truths about what we mean when we say "consciousness"? If so, can we go further and use consciousness to actually fully enumerate all the interactions necessary for it or all the implications of it from within a conscious system?