This question is about whether the definition of weighted Non-deterministic Finite state Automata (NFAs) excludes the possibility of infinitely many transitions.
The definition of Finite State Automata implies finitely many states, with a finite alphabet for possible transitions. However, in a weighted NFA, it is possible to have multiple arcs between the same two states with the same transition symbol but with different weights.
Now my question is: Could one define a NFA with two states and infinitely many transitions between them with the same symbol, e.g. one for each natural number?
The standard definition, as given in Wikipedia, in [1] or elsewhere, is that weighted non-deterministic finite automata only have finitely many transitions.
If you wanted to change the definition to allow infinitely many transitions, you would need infinite sums to define the behaviour of your automaton. This can be done in some semirings, like $({\cal P}{\Bbb N}), \cup, \cap)$, but not in every semiring.
Moreover, if you have two transitions, say $p \xrightarrow{a \mid x} q$ and $p \xrightarrow{a \mid y} q$, replacing them with the transition $p \xrightarrow{a \mid x+y} q$ gives you a weighted automaton with the same behaviour. Thus, without loss of generality, you may assume that for all $p$, $a$ and $q$, there is at most one transition of the form $p \xrightarrow{a \mid z} q$.
[1] M. Droste and D. Kuske, Weighted automata, Chap. 4 in Handbook of automata theory Vol. I. Theoretical foundations, 113-150, EMS Press, Berlin, (2021).