# How to denote enumerating 1, 2, …, n?

What is the best way to denote it?

1. $\forall i\in\{1,\dots,n\}: P(i)$;
2. $\forall i=1,\dots,n: P(i)$;
3. $\forall i=\overline{1,n}: P(i)$;
4. $P(i)$ for $i=1,\dots,n$;
5. ...
• I find it strange that there is no uniform notation for this common set. Another notation I have seen is $\underline{n}$. – copper.hat Sep 11 '12 at 15:29
• I have also seen $[1,n]$ on this site and $[n]$ elsewhere. – Ross Millikan Sep 11 '12 at 15:38
• Dem'yanov & Malozemov use $[1:n]$ in "Introduction to minimax". – copper.hat Sep 11 '12 at 15:44
• I use $(\forall i\in \mathbb{N}_n)(P(i))$. However in your notations, 1 and 4 are the best. 2,3 are not acceptable. – user59671 Feb 12 '13 at 14:46
• $i \gets$ enumerate $([1, n])$, and then let's invent a new math symbol for enumberate. – 象嘉道 Jan 3 '19 at 20:39

Use "$P(i)$ for $i=1,...,n$" to refer to the property being true in those cases. Using existential and universal quantifiers in ordinary mathematical prose (as opposed to formal logic text) is ugly.
I don't know if I've ever seen the notation $\overline{1, n}$, so I wouldn't use that :) But, maybe others know it, so maybe it's okay. Other than that, I wouldn't use $\forall$ or $:$ ever and most of my experience with such notation is in the class where you first learn it. I would use words but the basic forms of your 1, 2, and 4 seem pretty good.
$P(i)$ is true for all $i$ such that $1 \leq i \leq n$.
$P(i)$ is true for $i = 1, \ldots, n$.
For $i = 1, \ldots, n$, $P(i)$ is true.