Wikipedia defines as follows:
A basic definition of a martingale is a discrete-time stochastic process that satisfies for any time $n$:
$$\mathbf{E} ( \vert X_n \vert ) < \infty $$
$$\mathbf{E} (X_{n+1}\mid X_1,\ldots,X_n)=X_n$$
The second formula seems to indicate that the only thing that matters for the expectation is the previous value. Can we not just write:
$$\mathbf{E} (X_{n+1} \mid X_n)=X_n$$
Or is this somehow not equivalent? If they are not equivalent what is a counter example? It seems by not using $X_1,\ldots,X_n$ on the right hand side we are already indicating that they don't matter.
Let $Y_1,Y_2$ be independent with standard normal distribution. Let $X_1=Y_1,X_2=Y_1+Y_2,X_n=2Y_1$ for $n \geq 3$. Then $E(X_{n+1} |X_n)=X_n$ for all $n$. [For $n=2$ write $2Y_1$ as $(Y_1+Y_2)+(Y_1-Y_2)$ and use the well know fact that $Y_1+Y_2$ and $Y_1-Y_2$ are independent]. However $E(X_3|X_1,X_2)=2Y_1\neq X_2$.