"If there's a collection of third-order axioms that characterizes a model, there's a collection of second-order axioms that characterizes the same model. Once you make the jump to second-order logic, you're done - so far as anyone knows (so far as I know) there's nothing more powerful than second-order logic in terms of which models it can characterize."

You clearly can state Continuum Hypothesis in the higher order logic, while a 2nd order formulation seems elusive. Are you sure about it?