Transfinite Extension of the Mu-Calculus

In [1] Bradfield found a link between finite differences formed by Σ$^{\rm 0}_{\rm 2}$ sets and the mu-arithmetic introduced by Lubarski [7]. We extend this approach into the transfinite: in allowing countable disjunctions we show that this kind of extended mu-calculus matches neatly to the transfinite difference hierarchy of Σ$^{\rm 0}_{\rm 2}$ sets. The difference hierarchy is intimately related to parity games. When passing to infinitely many priorities, it might not longer be true that there is a positional winning strategy. However, if such games are derived from the difference hierarchy, this property still holds true.