The problem with cladistics is that it denies what it aims at

The basic idea of cladistics is that:

if you comprehend an entity (eg, yourself) yesterday as the ancestor of the same entity today, then the ancestor and the entity today can be said to form a group, which cladistics calls a "natural group", or a "clade". Cladistics ONLY acknowledges such groups.

As a corollary, cladistics does not acknowledge niether single entities nor categories.

The fundamental problem with this idea is that:

the facts that "clade" in practice is a category (ie, as the set of clades), and that cladistics denies all categories, mean that cladistics can't find a consistent set of "clades" (ie, a consistent category of clades) simply because it actually denies it (by denying categories).

The fundamental problem with cladistics is thus that it denies what it aims at (or aims at something it denies).

Isn't it a brilliant spin? (A pseudo-scientific approach that can't find a consistent solution per definition. Guaranteed life-long support and career in the academy).

Cladists have really stuck in an infinite loop

The fact that classification is orthogonal, ie, that every single class contains at least two classes, means that it has to be arranged orthogonally, ie, classifying entities into categories of classes, as in the Linnean system, to avoid the inherent contradiction of an orthogonality (see Russell's paradox).

However, such orthogonal system of classification is ambiguous in relation to the classified per definition, since it actually consists of two orthogonal classifications.

The approach in biological systematics called "cladistics" simply ignores this fact (ie, that classification is orthogonal) instead entering the inherent contradiction of an orthogonality (see Russell's paradox) while asserting (actually defining) that it, ie, the contradiction, indeed is real (ie, can be found). In this approach, ie, assuming that the contradiction is real, it is actually an infinite recursion, ie, an infinite loop. But, how can cladists possibly understand that they have entered an infinite loop when they don't understand that classification is orthogonal in the first place (actually not even that they classify)? No, they have indeed stuck in this infinite loop.

On classification of dichotomously propagating (bifurcating) processes

Dichotomously propagating  (bifurcating) processes can't be classified unambiguously. This does not, however, only concern this kind of process, but moreover all kinds of processes (ie, processes in some generic sense). The reason is that kinds of processes actually are classes (of processes), and classification of classes is ultimately contradictory, since classification is orthogonal. 

It means that kinds of processes are inconsistent (ie, contradictory) entities, and thus that classification into such entities is infinitely recursive. There simply is no consistent solution to any such classification.

This fact may appear counter-intuitive to some of us, but this appearance is only due to that those of us are not aware of that typification is classification. Those of us simply classify (ie, typify) without being aware of that it is what they do. Also illustrating a dichotomously propagating (bifurcating) process with a line graph is typification, since such graphs is a class. Understanding such illustrations consistently is a science on its own (ie, graph theory in mathematics), see "Are node-based and stem-based clades equivalent? Insights from graph theory" by Jeremy Martin et al. 2011.     

On the limits for classification (and the practical impossibility of cladistic classification)

The fact that classification is orthogonal, ie, that every single class contains several classes within every single set of entities (eg, the class "primates" and all different classes of primates), and that finite class thus is orthogonal to infinite class, means that classification is fundamentally contradictory between single and several, since orthogonalities are contradictory between single and several per definition. Cladistic classification accepts this contradiction, ie, accepts being contradictory, by equalizing the concepts finite class and infinite class.

The only way to avoid this orthogonal contradiction, ie, to achieve consistency, in practical classification is to use an orthogonal system of classification, classifying entities into categories of classes, like the Linnean system, in order to thereby keep finite classes and infinite classes (ie, the orthogonal concepts finite class and infinite class) consistently apart. Such orthogonal system of classification is, however, ambiguous in relation to reality per definition, simply by keeping the orthogonal concepts finite class and infinite class) consistently apart.

It means that classification can only be either internally contradictory or ambiguous in relation to reality. The fundamental reason for this impossibility to achieve unambiguity is that classes can't be (and thus aren't) real per definition, but can only be (and thus are) an artificial invention.

How to cure a cladist

If you believe that your classification is true (as cladists do), then you is contradictory, because classification is contradictory, which Bertrand Russell also demonstrated in 1901 with Russell's paradox.

It is thus not your belief (ie, that your classification is true) that is wrong, but what you believe in (ie, classification) that is. Your error does thus not reside in your conclusion (ie, that your classification is correct), but in your assumption (ie, that classification can be true).

If you end up in this belief (ie, cladistics), you thus have to scrutinize your assumptions.

Why should biological systematics accept a premise that is impossible?

If the class clade indeed is consistent, then there must be single entities (ie, single biological species) that can be divided into clades. Exactly how cladists mean that such division is possible remains to be explained. Until cladists provide an explanation of this impossibility, their approach thus ought to be discarded. Why should biological systematics accept a premise that is impossible?