Q:

Is there a concise way to notate 'There are exactly 482 x, such that Px...' in logical notation?

My prof has taught us that we can express the proposition $⟦$there are exactly two entities characterized by $P$$⟧$ thus:

That proposition looks verbose, despite the fact that it references just two entities. It seems impractical to use such a formula to express a proposition that references a great number of entities (e.g. 482). Is there a more concise way of conveying the number of entities that a proposition references?

A:

The standard abbreviation for "there are at least $482$ objects $x$ satisfying $\phi(x)$" is
$$
(\exists^{482} x)\phi(x)
$$
The extra quantifiers of the form $(\exists^{n} x)$ can either be defined directly in the metatheory, or viewed as abbreviations for longer formulas.
So you could write
$$
(\exists^{482} x)\phi(x) \land \lnot(\exists^{483} x)\phi(x).
$$
There is also a notation $(\exists ! x)\phi(x)$, which means there is a unique $x$ satisfying $\phi(x)$. I have never seen the notation $(\exists !^{482} x)\phi(x)$, but you could certainly define it in your prose.
Although I have seen this notation less, an SEP article suggests
$$
(\exists_{\geq 482} x)\phi(x) 
$$
and
$$
(\exists_{=482} x)\phi(x)$$

