User blog comment:B1mb0w/Rule 2B/@comment-27513631-20160206233504/@comment-27513631-20160207000140

The definition of $$\zeta_{\alpha+1}$$ is universally agreed to be $$\zeta_{\alpha+1} = \sup\{\zeta_\alpha+1,\varepsilon_{\zeta_\alpha+1},\varepsilon_{\varepsilon_{\zeta_\alpha+1}},\cdots\}$$ i.e. the $$\zeta$$ function enumerates the fixed points of the $$\varepsilon$$ function. This is far beyond the scope of ordinal exponentiation, and you get a whole new kettle of fish to deal with if you try to define generalised ordinal hyperoperators.