Pure birth process with constant rate — recognising a Poisson process

Suppose $\{X(t),t\ge0\}$ is a pure birth process with birth rate $\lambda_i=\lambda$ for all $i\in S$ and $X(0)=0$. The following are four statements: (I) $\{X(t),t\ge0\}$ is a process with stationary and independent increments. (II) $E(X(t))=e^{\lambda t}$. (III) $X(t)$ follows Poisson $P(\lambda t)$ distribution for every fixed $t$. (IV) $X(0)$ cannot be 0 in a pure birth process.