Is O(n) a proper class or a set?

Question

Is $O(n)$ as the collection of all functions that are bounded above by $n$ a proper class or just a set?

What about $O(\infty)$?

Answer 1

Since $\Bbb R$ is a set, we know that $\Bbb{R\times R}$ is a set, so $\mathcal P(\mathbb{R\times R})$ is a set.

Therefore the collection of all functions from $\Bbb R$ to itself is a set. In particular any definable subcollection of a set is a set. For example, all the functions which are $O(n)$ or otherwise.

Answer 2

If you are thinking of functions $f:\mathbb{N}\to\mathbb{R}$, you have a proper set.

Answer 3

There are multiple possible ways to define it, but on reasonable way to define it is like so:

$O(g(n))$ is the set of all functions $f:\mathbb{R}\to\mathbb R$ where there exist $k, N\in\mathbb R$ such that whenever $n>N$, $f(n)<k g(n)$. (i.e. $f$ is eventually bounded above by $g$)

That means $O(n)$ is the set of functions that are bounded above by the function $g(n)=n$.

$O(\infty)$ would be the set of all functions $\mathbb R \to \mathbb R$, since everything is bounded above by $\infty$.