User blog comment:Emlightened/Little Bigeddon/@comment-24061664-20170117003234/@comment-1605058-20170117162700

To the second question: the formula is in the language of set theory, there is no + symbol there.

To the first question: I've just realized Emli has probably messed up the indices, and it should be that if there are finitely many (n,k+1) formulas, there are finitely many (n+1,k). So once we show there are finitely many (0,n+1), then there are finitely many (1,n), (2,n-1), ..., (n,1).

As for why (0,n+1) is finite, we just have to note that the only formulas we can have without quantifiers are \(x=y,x\in y\) (where \(x,y\) are some free variables) and whatever we get from there using conjunction, disjunction and negation. It's not difficult to see there are only finitely many inequivalent ones there.