Why is $\left\{ n\in \mathbb{N}:n^2 \right\}$ nonsense, $ $ but $\left\{ n^2: n\in \mathbb{N} \right\}$ correct? Why is $\left\{ n\in \mathbb{N}:n^2   \right\}$ nonsense,  $ $ but $\left\{ n^2: n\in \mathbb{N}   \right\}$ correct?
From my understanding, $\left\{ n\in \mathbb{N}:n^2   \right\}$ should be read as:
"The set of natural numbers such that each natural number is multiplied to itself."
What's wrong with this?
Why is it not equivalent to $\left\{ n^2: n\in \mathbb{N}   \right\}$? $ $ which is read as:
"The set of $n^2$'s such that $n$ is a natural number." 

To confirm what everybody else has said, after the colon, you need a sentence that may or may not be true. Since there’s no verb in what follows the colon, your formulation is bad grammar.
