(i.e., each hyperbranch is in one and only one galaxy).

The [*principal galaxy*]{} $\Gamma_{0}$ of $^{*}\!G$ is that unique galaxy, each of whose hypernodes is limitedly distant from some standard hypernode (and therefore from all standard hypernodes). All the nodes in $G$ will be (i.e., can be identified with) standard hypernodes in $\Gamma_{0}$, but there may be nonstandard hypernodes in $\Gamma_{0}$ as well. The following examples illustrate this point.

[**Example 3.2.**]{} Consider the endless (i.e., two-way infinite) path: $$P\;=\;\langle\ldots,x_{-1},b_{-1},x_{0},b_{0},x_{1},b_{1}\ldots\rangle$$ with nodes $x_{k}$ and branches $b_{k}$, $k\in{Z \kern -7.5pt Z}$, ${Z \kern -7.5pt Z}$ being the set of integers. The enlargement $^{*}\!P$ of $P$ has hypernodes, each being represented by $[x_{n_{k}}]$ where $\langle k_{n}\rangle$ is some sequence of integers. Each hyperbranch is represented by $[\{x_{k_{n}},x_{k_{n}+1}]$. The nodal galaxies are infinitely many because they correspond bijectively with the galaxies of the enlargement $^{*}\!{Z \kern -7.5pt Z}$ of ${Z \kern -7.5pt Z}$. Moreover, the principal galaxy $\Gamma_{0}$ of $^{*}\!P$ has only standard hypernodes and in fact is (i.e., can be identified with) $P$ itself. Also, every galaxy is graphically isomorphic to $\Gamma_{0}$ and therefore to every other galaxy. $\Box$

[**Example 3.3.**]{} Now, consider a one-ended path: $$T\;=\;\langle x_{0},b_{0},x_{1},b_{1},x_{2},b_{2},\ldots\rangle$$ Each hypernode in the enlargement $^{*}\!T$ of $T$ is represented by $[x_{k_{n}}]$, where $\langle k_{n}\rangle$ is some sequence of natural numbers. Thus, $^{*}\!T$ has a hypernode set $^{*}\!X$ that can be identified with the set $^{*}\!{I \kern -4.5pt N}$ of hypernaturals. Hence, $^{*}\!T$ has an infinitely of galaxies, too. The principal galaxy $\Gamma_{0}$ of $^{*}\!T$ is the one-ended path $T$. However, any hypernode ${\bf x}=[x_{k_{n}}]$ in a galaxy $\Gamma$ different from $\Gamma_{0}$ will be such that, for every $m\in{I \kern -4.5pt N}$, $\{n\!: k_{n}>m\}\in{\