How to interpret the statement $\forall z. \exists u \forall x. (\exists y. x \in y$ AND $y \in z)$ IFF $x \in u.$?

Question

This is one of the ZFC Axioms. It says:

Union: The union, u, of a collection, z, of sets is also a set.

I interpret the statement in this way: For all z, there exists u for which for all x, there exists y such that x belongs to y AND y belongs to z, if and only if x belongs to u.

So this statement means if x is in both y and z, then it will be in u too. Since y is a set, making a union of all y's will have all the non repeating elements from those y's in u. Thus, u is a set.

I am not sure if my interpretation is correct or there is more to the statement. I don't know how to make a logical statement after reading the union axiom.

Answer 1

The Union axiom :

$∀z ∃u ∀x [x ∈ u ↔ ∃y(x ∈ y \text { and } y \in z)]$

asserts that for any given set $z$ there is a set $u$ which has as members all of the members of all of the members of $z$.

This does not mean : "if $x$ is in both $y$ and $z$, then it will be in $u$ too."

The set $u$ is the set of the elements of the elements of $z$; thus : "if $x$ is in $y$ which is in $z$, then it will be in $u$."

Answer 2

Your interpretation is incorrect in two ways:

When you say "$x$ is in both $y$ and $z$, you don't realize that $y$ and $z$ lie at different layers of sets: $z$ is a collection of sets, and $y$ is one of them.

Secondly, the statement is an equivalence, and your "interpretation" goes one way only.

The correct interpretation is "there is a set $u$ such that for any object $x$, $x$ lies in $u$ if and only if $x$ lies in at least one set $y$ from the collection $z$".

Answer 3

I'll give you an example and apply the axiom on it, and I hope this'll help.

$Z= \{\{1,2\}, \{2,3,4\}, \{5,6\}\}$ (a collection of sets)

$U=\{1,2\}\cup \{2,3,4\}\cup \{5,6\}=\{1,2,3,4,5,6\}$ (the union of the collection of sets, that is also a set)

Now we can say $\{1,2\} \in Z$, and we can say $2\in U$ or $3\in U$ ... and note that we don't say $2\in Z$ (because somehow $2$ is an element, so it can belong to a set of elements, but $Z$ is a set of sets, and that is why we can say that $\{1,2\} \in Z$)