There is no rule that states that a system should be consistent.
We impose the criterion of consistency on our logical/mathematical systems because reality seem to be perfectly consistent itself, and we have experience that one cannot be successful unless he respects that. He who neglects logic, shall logically perish.
Maybe paradoxes actually exist outside our arbitrary reality. But we may never know.
OP may call numbers generated by such expressions paradoxical numbers.
They may exist, they can be assigned symbols, but they are useless in this reality.