Non-classical foundations, violently non-set-theoretic, perhaps could cope with this while allowing numbers greater than six. Pretend there is a non-commutativ

Non-classical foundations, violently non-set-theoretic, perhaps could cope with this while allowing numbers greater than six. Pretend there is a non-commutative set theory where observations of membership do not commute akin to quantum mechanical systems. Perhaps it would be possible for a model of such a theory to have all finite sets as valid states, but sets of cardinality exactly 6 are forbidden. Just extremely sketchy hand waving but it might make sense. Certainly classical mathematics is out the window.