I was reading the proof given by Blass that the Existence of bases implies the axiom of choice. At the beginning of the proof he presents in that paper, the following is written:
Adjoin all the elements of $X=\bigcup_{i\in I}X_i$ as indeterminates to some (arbitrary) field $k$, obtaining the field $k(X)$ of rational functions of the "variables" in $X$. For each $i\in I$, we define the $i$-degree of a monomial to be the sum of the exponents of members of $X_i$ in that monomial.
Here $\{X_i\mid i\in I\}$ is a family of non empty sets that is pairwise disjoint.
My question is: If $X$ is an arbitrary set, with no clear algebraic structure defined, how can we speak of exponents of elements of $X$?
I am aware that we can form the ring $k(X)$ of polynomials in $X$ whenever $X$ has some operation defined, since exponentiation can be seen as a repeated application of the operation to the same element in this case. But I'm struggling to make sense of the definition in general. What would $x^n$ mean if $x\in X$ was... a set itself?
This is something that I have encountered previously when I saw linear combinations of simplices in algebraic topology. But every time I come across this concept I get more confused. My attempt to understand this was to consider each polynomial as a finite sequence, with the coefficients of the polynomial being the terms of the sequence, but what if the index set $I$ was uncountable? and what would it mean to multiply two variables $x_ix_j$ in $X$?
Also, is the concept of "free groups" related to this? In this article, the group word looks similar to raising elements of $X$ to some power.
Any help is gladly appreciated.