Question about how to interpret arbitrary elements I have a question about how to interpret arbitrary elements.
When we say:

*

*"let $x$ be an arbitrary integer"


*"let $x$ be any integer"


*"let $x$ be an integer"
at the beginning of proofs, are we saying:

*

*"let the symbol $x$ represent one of the integers, although it is not clear which specific integer the symbol $x$ represents".

If not, please let me know how I should think about the above statements.
Thank you for your time.

Edit:
I had a follow up question to the one I asked earlier. What do mathematicians mean by “arbitrary element” or “any element”. I can understand when mathematicians say “let $x$ be $4$” as this means “the symbol $x$ represents $4$”. Could someone please elaborate on why the specific terms “arbitrary” and “any” are used?
 A: Your statement is somewhat correct, but maybe not that useful.
Consider a statement like the following, which we'll call Statement 1:

Let $x$ be an integer. The number $4x+3$ is odd.

The power of this statement is its universality: it applies to every integer $x$. If I wanted, I could say Statement 2:

If $x=-26$, then $4x+3$ is odd.

This is a less general statement that only applies to a single integer $x$. I could also say Statement 3:

If $x=55$, then $4x+3$ is odd.

The power of a construction like Statement 1 is that it implies Statements 2 and 3: once we've proven it, we can use it for any integer we'd like. Statement 1 is not intended to be used for a specific value of $x$ that we decided not to name -- it can be used for any integer $x$ we want.

A side-note is that we can also apply such a statement to other "generic" integers. Consider Statement 4:

If $t$ is a positive integer, then $2^{t-1}$ is an integer.

We can combine the two to say that, if $t$ is a positive integer, then $2^{t-1}$ is an integer, and so if we set $x=2^{t-1}$ in Statement 1, we get that $4\cdot 2^{t-1}+3$ is an integer. So, Statement 4 and Statement 1 can be combined to give another general statement:

If $t$ is a positive integer, then $2^{t+1}+3$ is odd.

If we only cared about Statement 1 for a single integer $x$, doing something like the above wouldn't really be possible.
