# When should I make a colon in a statement?

As an example, take a look at the Cauchy's convergence test:

$\forall_{\varepsilon>0} \exists_{n_0\in\mathbb{N}} \forall_{m,n \geq n_0} \colon \left|a_m-a_n \right|<\varepsilon$

or

$\forall_{\varepsilon>0} \colon \exists_{n_0\in\mathbb{N}} \colon \forall_{m,n \geq n_0} \colon \left|a_m-a_n \right|<\varepsilon$

or

$\forall_{\varepsilon>0} \exists_{n_0\in\mathbb{N}} \forall_{m,n \geq n_0} \left|a_m-a_n \right|<\varepsilon$

Is there a difference in the meaning of these three statements? (If yes, please explain the difference)

When should I make a colon in a mathematical statement and when not?

(I've found the first and the second in Wikipedia)

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I wouldn’t write any of them. My preference is for $$\forall\epsilon>0\exists n_0\in\Bbb N\forall m,n\ge n_0\Big(|a_m-a_n|<\epsilon\Big)\;.$$ A distant second choice is $$(\forall\epsilon>0)(\exists n_0\in\Bbb N)(\forall m,n\ge n_0)|a_m-a_n|<\epsilon\;.$$ The only place where I use a colon in any sense related to this usage is in the definition of a set: $\{x\in S:\varphi(x)\}$. –  Brian M. Scott Aug 26 '12 at 14:39
–  Henning Makholm Aug 26 '12 at 14:50

## 1 Answer

It's just a matter of punctuation style. There is no mathematical meaning hidden in whether a colon is written or not.

Some authors prefer colons after every quantifier. Other omit them when the subformula itself starts with a quantifier. Yet other authors never use them.

Some authors use a dot instead of a colon. Some insist on parentheses around the $(\forall \epsilon > 0)$, or round the subformula, but rarely around both. Some will parenthesize the subformula except when it starts with a quantifier of its own, or depending on how tightly its top connective binds.

Some people (such as yours truly) do whichever things on alternate weeks, depending on the weather and the phase of the moon.

It's all just typography, and carries no logical content, as long as it's clear which parts of the formula each quantifier ranges over. (Beware that there are differing conventions about that, in the styles that don't insist on parentheses around the subformulas. So $\forall x:\phi\to\psi$ could mean either $(\forall x:\phi)\to\psi$ or $\forall x:(\phi\to\psi)$ depending on the exact conventions followed by the author).

In any case, note that writing the bound variable as a subscript to the quantifier, as you do in the question, is somewhat nonstandard.

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Thanks for mentioning that writing the bound variable as a subscript to the quantifier is nonstandard. The exercise instructor at my analysis course does it and I think it makes the expression easier to read. But its good to know that it's not the standard. And also thank you very much for the link you've provided in the comment. –  moose Aug 26 '12 at 16:37