Actually, I don't think I've seensuperscripts2 and greater. Usually, the superscript is just a mark whose numerical value is not important. Superscript 0 is used to denote classes of first-order formulas, and superscript 1 is used for second-order formulas.

In contrast, thesubscriptn refers to n alternating groups of similar quantifiers. So if P(x,y) and Q(X,Y) are quantifier-free formulas, then is a -formula (lowercase variables range over numbers), while is a -formula (uppercase variables range over sets of numbers).

See arithmetical hierarchy and analytical hierarchy.