Basic question on logic substitution Wikipedia explains that substitution is a function $\sigma : V \rightarrow T$ that maps all variables $x_i$ to the corresponding term $t_i$ for some natural number $i$. Additionaly, according to the article, some authors state that $\sigma(x)=x$. What does that mean? That the substitution function $\sigma$ is an identity function, in which every replaced variable $x$ is equal to the replacing term $\sigma(x)$? That interpretation does not seem quite right for me, since some paragraphs later, the text talks about invertible and not invertible substitution functions. If the substitution function $\sigma$ is an identity function, then, if $card(T) \leq card(V)$, $\sigma$ is surjective and injective and, hence, always invertible.
 A: Here's the full quote from that paragraph of Wikipedia for reference

In first-order logic, a substitution is a total mapping $\sigma : V \to T$ from variables to terms; many, but not all, authors
additionally require that $\sigma(x) = x$ for all but finitely many
variables $x$.

This means that $\sigma$ leaves all but finitely many variables the same when transforming one well-formed formula into another. "$\sigma(x)=x$" is not meant to imply that all variables $x$ are sent to themselves by $\sigma$, but rather that some of them (specifically cofinitely many) are.
The idea here is that a substitution, when written in the set of pairs notation shown below with fixed points omitted, should be finite.
$$ \{ (x_1, t_1), (x_2, t_2), \cdots ,(x_7, t_7) \} \;\; \text{completely describes $\sigma$ and is finite} $$
We could have chosen to define $\sigma$ differently, with a domain of $V'$, a finite subset of $V$. In that case, $\sigma$ when applied to a formula would only replace variables in $V'$. Variables in $V\setminus V'$ would be untouched.
