\cdots\},$$ is an obvious example of non-chained $g$-tower whenever $(\mathfrak{C}_{\sigma})$ is neither contained in, nor contains, any member of the $\mathfrak{C}_{\alpha}$-chain. Hausdorff’s chain theorem now yields the fixed-point $\widetilde{\mathfrak{C}_{\leftarrow}}\,\in\widetilde{\mathfrak{X}}$ $$\sup_{\widetilde{\mathfrak{C}_{\textrm{T}}}}(\widetilde{\mathfrak{C}_{\textrm{T}}})=\widetilde{\mathfrak{C}_{\leftarrow}}=\{\{ S_{\alpha}\},\{ S_{\alpha},S_{\beta}\},\{ S_{\alpha},S_{\beta},S_{\gamma}\},\cdots\}=g(\widetilde{\mathfrak{C}_{\leftarrow}})\in\widetilde{\mathfrak{C}_{\textrm{T}}}\subseteq\mathcal{P}^{3}(X)$$ as a maximal *totally ordered* $g$-chained towered subset of $X$ that is, by Zorn’s lemma, a maximal element of the base level subset $\mathfrak{S}$ of $\mathcal{P}^{2}(X)$. $\widetilde{\mathfrak{C}_{\leftarrow}}$ is a *chained principal ultrafilter on* $(\mathcal{P}(X),\subseteq)$ *generated by the filter-base $\{\{ S_{\alpha}\}\}$* *at $S_{\alpha}$*, while $\mathcal{S}_{+}=\{ S_{\alpha},S_{\beta},S_{\gamma},\cdots\}\in\mathcal{P}^{2}(X)$ is an (non-principal) *ultrafilter on* $X$ — characterized by the property that any collection of subsets on $X$ with FIP (that is any filter subbase on $X$) must be contained the maximal set $\mathcal{S}_{+}$ having FIP — that is not a principal filter unless $\mathcal{S}_{\alpha}$ is a singleton set $\{ x_{\alpha}\}$. $\qquad\blacksquare$

[1.4]{} What emerges from these application of Zorn’s Lemma is the remarkable fact that *infinities (the dot-dot-dots) can be formally introduced as “limiting cases” of finite systems in a purely set-theoretic context* *without the need for topologies, metrics or convergences.* The significance of this observation will become clear from our discussions on filters and topology leading to Sec. 4.2 below. Also, the observation on the successive iterates of the power sets $\mathcal{P}(X)$ in the examples above was to suggest their anticipated role in the complex evolution of a dynamical system that is expected to p