Potential misunderstanding in reading Set Theory by Hrbacek

Solution 1:

Short answer: No, the author is correct.

The Axiom of Pairing helps you to define sets $\{A,B\}$ whenever $A,B$ are sets. Then, by the Axiom of Union, you can form the set $\bigcup\{A,B\}$, usually denoted $A\cup B$.

You can now continue this process to form any union of finitely many sets, say $A_1\cup\cdots\cup A_n$. However, using this strategy, you will not be able to make the jump to the union of infinitely many sets.

In fact, in order to form the union $\bigcup_{i\in I} A_i$, where $I$ is some indexing set, and all $A_i$ are just some arbitrary sets, you proceed as follows: Use the Axiom Schema of Replacement (see later in the book) to form the collection $\mathcal C=\{A_i\mid i\in I\}$, then form its union $\bigcup\mathcal C$ using the Axiom of Union, which is precisely what we mean by $\bigcup_{i\in I} A_i$. As $I$ could be infinite, the union could be as well.