\mathbb{Z}_{+}}$, $1/2$ is in $\mathcal{D}_{-}$ because if the graph of $f_{2^{m-1}}$ passes through $(1/2,0)$ for some $m$, then so do the graphs for all higher values . Therefore $[0]=\bigcap_{m=1}^{\infty}\{ i/2^{m-1}\!:i=0,1,\cdots,2^{m-1}\}$ is the equivalence class of $(f_{2^{m-1}})_{m=1}^{\infty}$ and this sequence converges to $[-1,1]$ on this set. Thus our extension $\textrm{Multi}(X)$ is distinct from the distributional extension of function spaces with respect to test functions, and is able to correctly generate the pathological behaviour of the limits that are so crucially vital in producing chaos.

**4. Discrete chaotic systems are maximally ill-posed**

The above ideas apply to the development of a criterion for chaos in discrete dynamical systems that is based on the limiting behaviour of the graphs of a sequence of functions $(f_{n})$ on $X,$ rather than on the values that the sequence generates as is customary. For the development of the maximality of ill-posedness criterion of chaos, we need to refresh ourselves with the following preliminaries.

***Resume Tutorial5: Axiom of Choice and Zorn’s Lemma***

Let us recall from the first part of this Tutorial that for nonempty subsets $(A_{\alpha})_{\alpha\in\mathbb{D}}$ of a nonempty set $X$, the Axiom of Choice ensures the existence of a set $A$ such that $A\bigcap A_{\alpha}$ consists of a single element for every $\alpha$. The choice axiom has far reaching consequences and a few equivalent statements, one of which the Zorn’s lemma that will be used immediately in the following, is the topic of this resumed Tutorial. The beauty of the Axiom, and of its equivalents, is that they assert the existence of mathematical objects that, in general, cannot be demonstrated and it is often believed that Zorn’s lemma is one of the most powerful tools that a mathematician has available to him that is “almost indispensable in many parts of modern pure mathematics” with significant applications in nearly all branches of contemporary mathematics. This “