Intensional and extensional set specification

There are various conventions for denoting sets, and the choice comes down to whether an author wants to emphases set properties, set elements or some other aspect.

Extensional and intensional specification are two key ways in which an object, a term refers to, may be declared.

A set may be declared semantically using intensional specification like "a set containing odd natural numbers".

This may be denoted by:

$\{x . x \text{ is an odd natural number}\}$

or

$\{x \in \mathbb{N} . x*2\}$

Extensional definition defines an object by explicitly enumerating every element for which the defining property holds.

Defining a set extensionally means listing its elements one-by-one between the braces, which is also called roster notation:

$\{2,4,6,8,10\}$

Since it's inconvenient (or impossible) to define large sets this way, it is allowed to abbreviate the list using ellipsis, in case of both infinite and finite sets. For example:

$\{1,2,3, \dots\}$

or

$\{1, 2, 3, \dots, 100\}$

When a large finite set is denoted extensionally using ellipses (like above), the last element is called a terminal set member.