How to interpret "let" in mathematics? Before explaining my issue, I wanted to first explain what things do make sense to me.
So, statements (used at the beginning of proofs) like "Suppose $x$ is an integer" or "Assume $x$ is an integer" make sense to me. The way I read them is like: "Let's just pretend that the symbol $x$ happens to represent an integer".
Statement (also used in proofs) like "Let $x$ be 3" or "Let $x$ equal 3" also make sense to me. The way I read them is like: "Let's just temporarily name 3 with the symbol $x$.
My issue comes with statements like "Let $x$ be an integer" or "Let $x$ $\in$ $\mathbb{Z}$". I really do not know how I should intuitively interpret such a statement. I can't interpret it in the same way as like "Let $x$ be 3" because there is not a specific object being assigned like 3. Should I interpret it like how I interpret "Suppose $x$ is an integer"?

EDIT:
So, from what I gathered from the responses, I think I understand now how I should think of "Let $x$ be an integer".
I could think of it as "Assume a newly created symbol x happens to represent an integer". However, this can cause issues as then "Let $x$ be an element of the empty set" is also completely valid.
Instead, I should not think of "Let $x$ be an integer" as an assumption, but an assignment/declaration, just like "Let $x$ be 3". One tangible way to think about this is to imagine myself assigning the newly created symbol $x$ to an integer chosen in secret by a friend. With this mindset, $x$ is not assumed to be an integer - it is an integer. It is just unknown to me what integer it is.
I hope this made sense to anybody with similar questions. If anyone thinks I have made an incorrect finding, please feel free to correct me.
 A: *

*‘Let’ assigns meaning  (definitions or restrictions or values) to variables or phrases or symbols.

*

*In “let $k$ be an integer and $x=2k$”, $x$ is an arbitrary even number (or: the representation or general specification of the even numbers).


*In “let $x=7$”, $x$ has been instantiated with a specific value.
(or: “Put $x=7.$”)




*The adjective ‘arbitrary’ is frequently tacit, and these sentences all mean the same:

*

*Let $x$ be an arbitrary real number and suppose that $P(x).$

*Let $x$ be real and suppose that $P(x).$

*Let $x\in\mathbb{R}$ and suppose that $P(x).$

*Let $x$ be an arbitrary real number such that $P(x).$

*Let $x$ be real such that $P(x).$

*Let $x\in\mathbb{R}$ such that $P(x).$



*‘Any’ versus ‘arbitrary’.
A: You should take "Let $x$ be an integer" as a synonym for "Assume that $x$ is an integer" or "Consider an integer $x$".
A: "Let" is usually used to introduce a new symbol along with an assumption about it ("Let $G$ be a group") or when defining notation ("Let $\mathcal P(S)$ denote the power set of $S$"). "Suppose" and "assume" are more conventional when introducing new assumptions using only existing symbols and notation ("Suppose $x^2<5$"). This isn't a rule, though - you can certainly introduce new symbols with an "assume" or "suppose".
I'd also say "suppose" is slightly more common when introducing a proposition that is "in doubt" and will shortly be refuted or discarded (e.g. with a proof by contradiction), whereas "assume" is more neutral in tone and is more common when introducing long-lasting assumptions or "background information".
The three words have the same technical meaning, in that we'd translate them into formal logic in the same way.
