Are there any rules concerning the variable symbols in first-order languages?

Question

I have read that one part of alphabet in first-order languages is an infinite collection of variables. Are there any rules about what the symbols of these variables should look like? One can often read that they're assigned lowercase letters from near the end of the Latin alphabet, often with numerical subscripts, e.g. $x_{353}$. However, what about variables like $a^{\rightarrow}$, a symbol of a vector, or Greek letters often used for angles in geometry? Variables in real world mathematics often use symbols very different from "lowercase letters from near the end of the Latin alphabet". So are there any rules about what should variable symbols look like, besides "different from symbols already defined as not being variables"?

Answer 1

Note that real-life math formulas are not what we learn to call well-formed-formulas of first-order languages. This follows already from the two-dimensional arrangement in things like $$\sum_{k=0}^n{n\choose k}x^k=(1+x)^n $$ That being said, in order to really specify a formal language strictly, you need to define your alphabet, including that countable set of variables, for example as one specific letter $x$ indexed by a natural number (note that this would make things like $x_{1728}$ a single symbol!); or as one of several specified letters, possibly followed by a finite sequence of apostrophes (so a symbol sequence such as $x''''$ would represent a variable syntactically).