User blog comment:Tetramur/Well-defined Oblivion number (I think)/@comment-35470197-20200106221111

No. Set theory needs infinitely many symbols if you are considering first order logic, because it is not finitely axiomisable. If you allow logic of any order, you need to specify the syntax. If you are considering the number of sorts of symbols, then every countable theory can be described by a single symbol. If you have infinitely many candidates for K(1), then the whole system is not well-defined because every number (in meta-theory above microlanguages which you are considering) can be defined by some system.