I am reading Robert André's book Axioms and Set Theory and, maybe starting at chapter 12, he uses the notation like $\mathbb{N}=\cup\{n \mid n \in \mathbb{N}\}$, where he puts a union symbol (∪) before the set builder notation. I do not know what it means, though I suspect it is a typo.
Hence, my question: what does $\cup\{x \mid \phi(x)\}$ mean?
Best Answer
In a set theory development of the natural numbers, what we do is we start with nothing but sets and then build structures within that mimic our knowledge of numbers and operations on them. The standard construction for the natural numbers is to set $$0:=\emptyset $$ and then inductively define each following natural number by the successor function $s(x)$ which gives the next number after $x$ as $$s(x+1)=x\cup \{x\}$$
So $0=\emptyset$ $1=\{0\}=\{\emptyset \}$, $2=1 \cup \{1\}=\{0\}\cup \{1\}=\{0,1\}$, etc.
Note in this construction the only objects are the empty set and sets built from the empty set, the numbers are just names we assign to certain configurations of the sets. In this configuration, it is true that any natural number is just the set of all the previous numbers
$$n=\{0,1,2,...,n-1\}$$
Thus it makes the sense to take the union of the numbers because numbers are just names for special sets. And when you take the union of all of them, you get all natural numbers.
Added: For your exercise, you just want to show the two sets are equal, so as usual, show they are subsets of each other. So show any natural number is in that union, and then show that any member of that union is in fact a natural number
Second addendum: Technically the author should have written
$$\bigcup_{x\in \mathbb N}x$$ but when the context of the index set is obvious, we often do abuse of notation and skip writing the index set and go straight to $$\bigcup x$$