Arnold's proof can be used to show that certain classes of functions are insufficient to express a quintic formula.

These classes can always safely include all single-valued continuous functions (you cannot even write the _quadratic_ formula in terms of arithmetic and single-valued continuous functions!), but also plenty of non-single-valued functions (e.g. the +-sqrt function which appears in the well-known quadratic formula).

Applying Arnold's proof to the class given by arithmetic and all complex nth root functions (also multivalued) gives the usual Abel-Ruffini theorem. But Arnold's proof applies to the class "all elm-expressible functions" without modification.