The language of mixed hypergraphs allows us to interpret a concept which is hidden in the notion of a natural number. This could be described as follows.
Any concept in our mind that can be modelled by a mixed hypergraph $\mathcal{H}=(X,\mathcal{C},\mathcal{D})$ is a result of interaction between two families of opposite constraints expressed by $\mathcal{C}$ and $\mathcal{D}$. If $\mathcal{H}$ is colorable, then we have a solution of this interaction. However, if $\mathcal{H}$ is uncolorable, then we have no solution, and the respective concept can be treated as “self-contradictory”. The “more uncolorable” $\mathcal{H}$ is, the more self-contradictory the respective concept is.
In real life any object is identical only to itself. Even if we have object A and object B which look identical (say, two points), they, in fact, are different because they have distinct positions in space. Counting may be viewed as a sequential naming of objects which are in fact pairwise different (= of different colors), but by definition are considered pairwise identical (= of a common color). Therefore, if $X$ is a set of such objects, then the counting of $X$ corresponds to a step-by-step construction of a mixed hypergraph $\mathcal{U}_n=(X,\binom{X}{2},\binom{X}{2})$. The concept of the natural number $n=|X|$ arises as a result of $\binom{n}{2}$ elementary contradictions (= “minimal uncolorabilities”) generated by pairwise comparisons.
As we have seen in Chapter 3, among all uncolorable mixed hypergraphs on $n$ vertices, $\mathcal{U}_n$ is the “most uncolorable” since each of its non-trivial induced subhypergraphs is uncolorable as well. Moreover, $\mathcal{U}_n$ is the unique such mixed hypergraph for every $n$. This means that whatever concept based on comparison we would have on $X$, the concept of the natural number $n$ is the most self-contradictory.
In fact, real counting begins when we start comparisons, i.e. with the number 2. Thus for each $n$, we have $n-1$ such steps which coincides with the vertex uncolorability number $\Omega_X(\mathcal{U}_n)$.