notations of sets

A set is a collection of objects of any kind. The most important thing is that elements (i.e. objects themselves ) of a set have been defined mathematically with no doubt.

(a) a set $A$ of integer numbers, whose elements are very great numbers.
(b) a set $A$ of integer numbers, whose elements are greater than $10$ .

You must have understood (a) is a insufficient definition of  a set. Because with (a) we are not able to decide whether the number $1,000,000$  should be in $A$ . (please recall Archimedean properties)

A set $A$  with the definition (b) is expressed by two ways, as are well known.

(1) $\left\{ 11,12,13,\cdots  \right\}$
(2) $\left\{ x | x>10, x\in\mathbb{Z}  \right\}$

(1) is said to be a extensional definition and (2) is connotative. For a finite set, if we adopt a extensional definition, we have to write all elements in principle. However, for a set whose elements are much more or infinite, as it is impossible to do, we can use  "$\cdots$ ". In that case, we have to leave no ambiguities for "$ \cdots $ " .

In a connotative definition $\left\{x | C(x) \right\}$ , $x$  is a variable number satisfied the conditions $C(x)$ . Of course, $\left\{x | C(y) \right\}$ is unacceptable.