Language usage: "for" vs. "for all" Every student of math learns about the universal quantifier "for all/each/every". On the other hand, in mathematical writing, it is fairly common to read e.g. "for $i=1, \ldots n$, let $\ldots$". I had long assumed that the "for" here is identical to "for each" (less commonly, it might more reasonably stand for the existential quantifier "for some"), but where the author is perhaps getting sloppy with their language.
Now, based on things I've encountered over time, I'm less sure about my initial understanding. (Something closely related is the usage of "for" as one of the standard looping structures in programming languages.) So I want to clear this up once and for all. Does the word "for" in this context have any meaning or usage separate from standing for the universal quantifier? To ask this in a different way, is there ever a situation where you can write "for $i=1,\ldots, n$ but not "for all $i=1,\ldots,n$"?
 A: No.  The use of "for" rather than "for all" here is stylistic; the second way simply sounds strange.
It is analogous to the standard use of "if" instead of "if and only if" when making definitions in mathematics, which is also something that people ask about (see here).  When we write "a set $G$ with a binary operation on it is called a group if [put axioms here]" the intended meaning is "... called a group if and only if..." but nobody writes the second version; it sounds very strange in English to write like that.
You might want to read here for a discussion of "any" vs. "all". That is a situation with some ambiguity for people learning mathematics, especially if English is not their native language.
A: Nope, they're the same. The clearest rephrasing in my opinion would be

for each $i\in\{1,...,n\}$, [stuff]

that is, explicitly say the "each" (or whatever) and shift from equality to elementhood. Another rephrasing would be

for each $i$, if $i=1$ or $i=2$ or ... or $i=n$ then [stuff],

which reveals the "disjunctive" nature of the notation "$i=1,...,n$."
