Is random walk on half-line a martingale?

Question

Let $X_n$ denote a random walk on $\mathbb Z^+$ starting at $0$. Is it a martingale?

In Probability with Martingales by David Williams on page 99 it is claimed that it is, but I cannot understand why. I am trying to check the definition and get the following $$ \mathsf E(X_{n+1}|X_1, \dots, X_n) = \begin{cases}1/2,& X_n = 0\\ 0, & X_n > 0\end{cases} $$ which is not zero, hence $X_n$ is not a martingale. (Or I am using a wrong definition of $X_n$...)

Answer 1

This is a typo, one should read "Let $X$ be a simple random walk on $\mathbb Z$".