I'm reading Paul R. Halmos' book on Naive Set Theory. Though I'm very fond of the book, I believe it could still provide more details during it's explanations; for, I do experience certain ambiguities while reading. I'm familiar with the Cartesian Product of exactly two sets, expressed using the notation $X \times Y$, if $X$ and $Y$ are the sets considered. Informally, also, I could define the Cartesian Product of exactly to sets to be all the possible ordered pairs with the first coordinate belonging to the first set, and the second coordinate to the second set.
Now, when Paul Halmos talks about Cross Products of families (say, chance) the cross product of the family $\{X_i\}_{i \in I} = \{\{a, b, c\}, \{d, e, f\}, \{g, h, i\}\}$ and $I = \{1, 2, 3\}$, what exactly is the extension of the so-defined set? Does this informally mean the set of all ordered triples (for the size of $I = 3$) such that the first coordinate belongs to the first set, the second to the second, and the third to the third? If so, how do we decide which set is to be called the first, second, or third? Just because I have so labeled them them $1, 2$ and $3$ doesn't set-theoretically denote them to be the first, second and third; perhaps, it does intuitively, but there's a leap that's huge, impossible even, between intuition and mathematical precision.
I could have also defined the index set to be ${a, b, c}$, and if it is to rely on alphabetical order, then I could ultimately just make the indexed set itself the indexing set. I seek elucidation on this matter, oh reader who must be more professional than me. Thank you in advance
The cartesian product of a family of sets indexed by some set $I$ is independent of any order you happen to impose on $I$. You can define it as the set of functions $f$ from $I$ to the union $\cup_{i \in I}X_i$ such that for all $i$, $f(i) \in X_i$ (though Halmos may not do this).
Formally, there are no tuples - no ordered pairs, or ordered triples. In practice, $I$ is often an initial segment of the natural numbers and it's convenient to think of the first, second, ... components. I suspect that's all Halmos needs.
What really matters when thinking about the cartesian product $P$ is how the projections behave. Those are the functions $\pi_i: P \to X_i$ that single out the "$i$th" component of the tuple you are imagining.
See https://en.wikipedia.org/wiki/Cartesian_product