$\text { The solution set for the inequality } 13 x-5<15 x+4<7 x+12 ; x \in W \text { is }$

The inequality $13x - 5 < 15x + 4 < 7x + 12$ can be split into two separate inequalities:

First inequality: $13x - 5 < 15x + 4$
$-5 - 4 < 15x - 13x$
$-9 < 2x$
$x > -4.5$

Second inequality: $15x + 4 < 7x + 12$
$15x - 7x < 12 - 4$
$8x < 8$
$x < 1$

Combining these, the solution set for $x$ is $-4.5 < x < 1$.

Since $x \in W$ (the set of whole numbers, $W = \{0, 1, 2, 3, \dots\}$), we look for integers in the interval $(-4.5, 1)$ that are also whole numbers.

The integers in the interval $(-4.5, 1)$ are $\{-4, -3, -2, -1, 0\}$.

Among these, only $0$ is a whole number. Therefore, the solution set is $\{0\}$.

Answer: $\{0\}$