My question is pretty basic. In Set-Theory we're used to the construction of new sets from the existing ones using the Axiom of Specification. But what if I'd like to construct a set of all ordinals less than some fixed ordinal?
E.g. Let's take e.g. $W(\omega_0) = \{ \beta \ | \ \beta < \omega_0\}$. Axiom of Specification requires one to construct a set out of an already existing set. Since we know that the collection of all ordinals is not a set, then how can one even build the above $W(\omega_0)$?
In other words, I have the $W(\omega_0) = \{ \beta \in \textbf{?} \ | \ \beta < \omega_0\}$ problem.
The set $W(\omega_0)$ that you want to define is the set $\omega_0$.
An ordinal is just ( as a set) the set of all its predecessors, cf.
$0=\emptyset$, $1 = \{0\} = \{\emptyset\}$, $2=\{0,1\} = \{\emptyset, \{\emptyset\}\}$, $\omega= \{0,1,2,3,\ldots,\}$ etc.
See the definition by von Neumann on the Wikipedia page, e.g. The axiom of foundation in ZF allows us to define ordinals as transitive sets that are linearly ordered by $\in$. It's quite elegant IMO.