How to define a set without set builder notation How do you define a set without using set builder notation? For example, let's say that I want to define set S as:
$$ S = \{ x \in \mathbb{N}\ \mid 0 < x < 5\} $$
Then $$ S = \{1, 2, 3, 4\} $$
However, suppose that I wanted to define S without set-builder notation, as below?
$$ \forall x \in \mathbb{N} (\ 0 < x < 5 \iff x \in S \ ) $$
Are these two statements equivalent?
 A: They are not equivalent. If $S'=\{1,2,3,4,5,6,7\}$ then we have :
$$\forall x \in \Bbb N \quad 0<x<5 \Rightarrow  x \in S'$$
but $S \neq S'$.
A: A fool-proof alternative would say
$$\forall x: (x\in \mathbb N \wedge 0<x<5) \Leftrightarrow x\in S$$
This rules out all $x \notin \mathbb N$, which your initial proposal did not and it also formulates it as equivalencies.

Note that always the following representations of an arbitrary set $A$ are equivalent:
$$A = \{x\in X: \phi(x)\}$$
and
$$\forall x': (x'\in X \wedge \phi(x')) \Leftrightarrow x'\in A$$
A: Set-builder notation can be taken as a contextually defined abbreviation. At least this is how I learned it.
We can take $y\in\{x:\phi\}$ as short for $\exists z(y\in z \wedge \forall x(x\in z\leftrightarrow \phi))$. An obvious variation works for $\{x:\phi\}\in y$ and for contexts involving "$=$". In principle, we don't usually need anything in the language of set theory but the machinery of FOL and $\in$, so pretty much anything else is a handy abbreviation.
A: Apart from errors in logic, what you have is a proposition which happens to be true for your set $S$. This is not the same thing as an expression for $S$. You can't, for instance, write:
$$3\in \left(\forall x \in \mathbb{N} (\ 0 < x < 5 \Rightarrow x \in S \ )\right)$$
Whereas you can write $3\in \{1,2,3,4\}$.
