The ellipsis "$...$" sometimes seems to be seen as a bit informal. Its use is often justified for cases "where the intention or meaning is clear". And of course, one can go arbitrarily far with nitpicking, intentionally misunderstanding a notation, or suggesting ambiguities. In many cases, the intention really is clear. But for me, it still looks like the writer was handwavingly saying: "Yeah, and so on, y'know what I mean". Usually, the ellipsis can easily be replaced with a more rigorous notation - often involving some sort of indexing over $\mathbb{N}$. And I wonder why this is often not done.
So my question is:
How acceptable is an ellipsis "$...$" in formal mathematics?
Of course, this does not refer to textbooks where the natural numbers are introduced as "$\{1,2,3,4,...\}$". It rather refers to mathematical research, or as a specific example: A paper about a proof where the correctness of the proof crucially depends on the right interpretation of an ellipsis, even if it is only used in a basic definition of something "trivial and obvious" like "$a_1 + ... + a_n$".
How far should one go with trying to avoid the use of the ellipsis, in order to not be confronted with the possible ambiguities or lack of rigour?
I found two questions that are related to this one:
- I've been told that the use of ellipsis in "$S = x_1 + x_2 + x_3 + x_4 + \dots$" is ambiguous and meaningless. Is it?
- More rigorous notation than "ellipsis" for "and so forth?"
They refer to a particular use of ellipsis "$...$", and how to replace it with a more rigorous notation. Further search reveals attempts to formalize the ellipsis - for example, Proofs About Lists Using Ellipsis (A. Bundy, J. Richardson) states
A notation often used in informal mathematical proofs is ellipsis (the dots in $a_1 + ... + a_n$)
...
The first problem in formalising ellipsis is its inherent ambiguity. The reader of a formula containing ellipsis has to induce a pattern from the expressions on either side of the dots. [...] One can try todisambiguate ellipsis by putting in more context [...] but some ambiguity will always remain. More importantly, it is hard to see how we can ensure that a “proof” is in fact a proof unless it can be expressed in an unambiguous internal representation
But this refers to a very specific context, and not to how acceptable the ellipsis is in proofs and definitions in general.
\ldots
, rather than...
. In the context of a sum, you should use\cdots
to produce $a_1+\cdots+a_n$. $\endgroup$