When we want to prove that some property is not expressible in certain logic, we often use EF-game or it's variants to show this result in finite model theory.
for example, we can show that connectivity is not FO-expressible by classic EF-game on finite structures. When we add fixed-point formulas, in particular Least fixed point or Inflationary fixed point, how we can show that some property is not expressible in this logic?
Is there a similar way ? If it is not; is it well studied?
