In my Caculus class we have derived the Real numbers by assuming that $\mathbb R$ is a field. That means that we have assumed that $+$ is commuative by postulating that $a + b = b + a$ for every $a,b\in\mathbb R$. We then showed that $\sum_{i=1}^\infty x_i$ is not commutative (e.g. consider $x_i = (-1)^i$). So we know that finite sums are commutative. That is, we postulated that $a+b=b+a$ for every $a,b\in\mathbb R$ and noticed that this only holds for FINITE sums.
When we introduced sets we made a similar assumptions for the union: we assumed that $A \cup B = B\cup A$ for sets $A,B$. However, $\bigcup_{i=1}^\infty X_i$ is in fact commutative. How do we actually know this?
In other words: In both cases (and one could additionally consider propositional calculus and $\vee$ which is commutative as well) we have assumed an equality which clearly holds only for a finite number (=2) elements. For the first case we could find a counter example to prove that it in fact only holds for a finite number of elements. For the second case there is none. But how do we know that? I mean from a logical perspective one should prove that starting with a finite number of elements it also holds for an arbitrary number of elements. Yet I have never seen a proof for that. Can someone explain me this contradiction?
We can't conclude from finite to infinite in general. In order to show that commutativity works for some specific operation, we need to prove that it holds for that specific operation using what we know about that specific operation.
Even the fact that the infinite operation makes sense isn't something we can show in general, and will have to rely on each specific case to see whether it does. Infinite sums usually do not make sense, although in some limited number of cases we can give it a consistent value. Union, by its nature, generalises from finite to infinite without issues.