Indexing Variables When picking variables it is common to use prime marks, as in $a, a',a'', ...$, and numbers $a_1, a_2, a_3, ...$. A third option is to use distinct letters $a, b, c, ...$. For example, some people write an exact sequence of $R$-modules as $0 \rightarrow N \rightarrow M \rightarrow L \rightarrow 0$, whereas some prefer to write $0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$, others $0 \rightarrow M_1 \rightarrow M_2 \rightarrow M_3 \rightarrow 0$.
Such conventions help people to read and write mathematics quickly. Readers of each proof must memorize these notations, and so which convention to pick is worth some thought. Hence I am collecting explanations as to why people prefer one of these conventions over the others.
For example, it may be relevant to note that the use of prime marks as in $a, a', a''$ seems to privilege $a$, while the use of different letters $a, b, c$ does not seem to do this. This would suggest that $a, a', a''$ be used in a situation where $a$ is the more essential variable.
A related interest I have is in arguments that one might prefer to index sequences starting from $0$ or $1$. Dijkstra gives an argument explaining why, after contemplation, many prefer to start with $0$:
https://www.cs.utexas.edu/users/EWD/transcriptions/EWD08xx/EWD831.html
That's interesting, but now I'm wondering about the three conventions above. So, are there reasons one might prefer one of these three conventions- prime marks, numbers, or distinct letters? 
 A: A few thoughts on each. I'm not a professional mathematician, just a learner who did a mathematically-heavy subject at university, but these are the main considerations that come to my mind:
Prime marks:


*

*difficult to read if you need more than about $3$ of them ($4$ at a push).  

*can be misread as indicating derivatives if there's calculus going on.  

*don't mix well with indices, especially when handwritten: eg $a'^2$ ends up needing to be $(a')^2$ for legibility, when it could just be $b^2$.  

*nice and concise otherwise.  

*I tend to expect $a'$ to be the result of doing something to $a$.


Subscripts:


*

*can get messy if attached to a symbol that appears very frequently, appears in an exponent, or is raised to a power.  

*can make an argument look more complicated than it really is, if they do get messy.

*can be numbered as high as you like without losing legibility.  

*can be in the form of expressions, e.g. $a_{n+1}$, giving more flexibility.  

*often emphasise that you're dealing with an arbitrarily large set of instances of something: e.g. coefficients in a series or terms in a sequence.


Letters:


*

*can be chosen for intuitive ease of understanding, especially ones that are conventional (like $t$ for time, or for the independent variable in a parametric equation).  

*are clearly distinct from each other. (Compare solving a set of equations in $x, y$ and $z$ with solving the same equations but in $x_1,x_2$ and $x_3$.)

*but if you need a lot of them, you soon have to start using unsuitable ones because the obvious ones aren't available.

*a few, like $e$, can cause at least momentary confusion because they're used as the usual name for something.


There are probably also considerations about ease of typesetting if the material is for publication—I've no experience of that, but the standard reference (in the UK) on copy-editing includes a long section on editing mathematics.
