As an actuarial student I would like to know why the force of mortality is used to model survival probabilies. It wouldn't be easier to model directly the shape of the survival function?
the force of mortality is defined as: $\mu_x=-\frac{S'(x)}{S(0)}$
while the relationship with the survival probability is, indeed, $S(t)=e^{- \int \limits_{0}^{t} \mu_s \; ds}$
For your understanding of formulas, $T_0$ is the random variable describing the lifetime of a newborn, $S(0)=1$ is the survival function that evaluated in $t=0$.