I am reading about ANOVA decomposition to understand how Sobol sensitivity analysis works. In one book, a generalised ANOVA decomposition of $f(\bf{x})$ is given as $$ f(\mathbf{x}) = f_0 + f_1(\mathbf{v}) + f_2(\mathbf{w}) + f_{12}(\mathbf{v}, \mathbf{w}) $$ where $\bf{x}$ is the set of all input factors, $\bf{v}$ is the subset of factors from $x_1$ to $x_t$, and $\bf{w}$ is the subset of remaining factors.
I am confused about the meaning behind the function subscripts, can anyone enlighten me on what their purpose is? My (probably naive) assumption is that the equation would have the same meaning if they weren't there.