Let the universe be $ \hat{V}$, which is constructed as:
$ \hat{V} := V_0 \cup V_1 \cup V_2\cup ... $
where
$V_0$ is the set of primary elements and
$V_{v+1} := V_v \cup P(V_v)$
So, in other words, the universe is pretty much the ordinary nonstandard-analysis universe.
My question is: How can this universe contain the $\sum$-function for an infinite series of primary elements?
While finite sets, pairs and unions of elements of $\hat{V}$ can be proven to be also in the set, once stuff goes to the infinite, the set doesn't necessarily contain them anymore. An example would be the infinite union $\bigcup_{i=0}^{\infty}V_i$, which is equal to $\hat{V}$, which is no item of itself.
Just to add some examples/optional characteristics for this universe:
Ordered pairs are defined as by the Kuratowski definition, i.e. $(a,b)_{k} := (a,k) :=\{\{a\},\{a,b\}\}$
$V_2$ already contains all ordered pairs $(a,b)\quad \text{where}\quad a,b\in V_0$ .
A binary relation $R$ is defined as a set of pairs. E.g. if $R\subset A\times B$ where $A,B\in \hat{V}$, the relation $R$ is defined as:
$R = \{(a,b) \in R : a\in A, b\in B\}$
Okay, I think I've got the answer. Instead of proofing that an infinite pair $(a_1,a_2,...,a_{\infty},a)$, where the $a_k$ are the addends and $a$ is the result of the infinite sum (which, written down like that already looks seriously wrong) is in the universe, I'm looking at the system of functions $f:\mathbb{N}\rightarrow V_0$ (the set of sets of two-pairs $(a,b)$ where $a\in\mathbb{N}$ and $b\in V_0$), and construct the $\sum$-function as a set of pairs $(f,z)$, where $f$ is an element of the system and $z\in V_0$.
