I'm given the following predicate that I'm trying to understand from a course example.
$\forall n \in\mathbb N : n^2 \ge n$
is the colon for such that correct?
I understand that it reads as "for all elements in the natural set such that the square of n is greater or equal to n"
Should it be $\forall n \in\mathbb N, n^2 \ge n$? "for all elements in the natural set the square of $n$ is greater or equal to $n$"
it makes more sense to me this way. But I'm not sure if my assumption is correct.
As always, you should defer to your professor on matters of convention. If he says to use a colon in such and such way, do it. That being said...
Usually you see a colon in set builder notation, ie $$\{n\in\mathbb{N} : n^2 \geq n\}$$ In which case you would read it as "(the set containing) all natural numbers such that $n$ squared is greater than or equal to $n$."
The colon is usually not used in propositions meant to be read, so you wouldn't use $$\forall n\in\mathbb{N}:n^2≥n$$ to mean "for all elements in the natural set such that the square of n is greater or equal to n... (insert rest of proposition)."
You can simply use a comma to mean "such that," but generally it makes your math much less understandable. For instance, "$\forall n\in \mathbb{N}, n^2 \geq n$" could mean "...for all $n$ such that $n^2 \geq n$" or it could mean "Each $n\in \mathbb{N}$ has the property that $n^2 \geq n$"
Shorthand should never sacrifice clarity, so I would recommend "$\forall n\in \mathbb{N}$ s.t. $n^2 \geq n$..." Basically,
You are correct. The punctuation in the examples \begin{gather}\forall n{\in}\mathbb{N}:n^2≥n\tag1\\\exists n{\in}\mathbb{N},\,n^2≥n\tag2\\\exists n {\in}\mathbb N:n^2 \ge n\\\forall n {\in}\mathbb N,\, n^2 \ge n\end{gather} are superfluous, primarily to supply typographical breathing space. In $(1)$ and $(2)$ in particular, the punctuation can be confusing (and break the flow of reading), as evidenced by your Question. In $(1),$ the colon is not to be read as "such that" (doing so renders the statement ungrammatical and incoherent); in $(2),$ the comma is similarly neither literal nor to be understood as a pause.
In more complex formalisations, to delimit quantifier scope, commas/colons/periods are sometimes used in place of parentheses. Because this usage of punctuation conflicts with the above, punctuating quantifiers potentially creates ambiguity and even misinterpretation.
Here's an example where adding commas creates confusion and ambiguity.
