The battle beween Linnean systematists and cladists concerns only one issue: whether cladification (i.e., classification into only clades) is contradictory or not.
The solution to this issue resides in understanding that the ultimate clade, i.e., The Tree of Life, is the set of all sets in set theory. This set was shown already by Cantor (1891) by his diagonal argument (and later by Russell’s paradox (1901), the first of Gödel’s incompleteness theorems (1931) and Turing’s answer to the Entscheidungsproblem (1937) to be an inconsistent notion. If S is the set of all sets then P(S) would at the same time be both bigger than S and a subset of S.
The founder of cladistics, Willi Hennig, was thus either late in, or out of phase with, science, when he stated the strange claim that only sets of all sets are “natural” groups. Such sets had thus been shown to be contradictory long before he stated this claim. If only such sets are “natural” groups, then such “natural” groups can’t be found per definition, since they are contradictory.
The battle beween Linnean systematists and cladistists (i.e., concerning the issue whether cladification (i.e., classification into only clades) is contradictory or not) was thus won by Linnean systematists long before cladistics was born. Cladification was shown to be contradictory long before Hennig stated his strange claim.
It means that cladification can’t reach a non-contradictory solution per definition. Instead, cladification is, in practice, a red herring (or a pink elephant). It is, in practice, doomed to an eternal wandering between different contradictory solutions.
Cladification is thus, in practice, a return alley in our search for perfection. The main road is, instead, a classificatory system of the Linnean kind, that is, an orthogonal system. Exactly how it ought to be devised to close up on perfection is an open question, but the fact that it is the main road is not at question.