Gödel's incompleteness theorems however, it is not possible to use first-order logic to prove any such particular axiomatic set theory is free from paradox. How sets are defined and set notation Mathematical texts commonly denote sets by capital letters in italic, such as , , . A set may also be called a collection or family, especially when its elements are themselves sets. Roster notation Roster or enumeration notation defines a set by listing its elements between curly brackets, separated by commas: In a set, all that matters is whether each element is in it or not, so the ordering of the elements in roster notation is irrelevant (in contrast, in a sequence, a tuple, or a permutation of a set, the ordering of the terms matters). For example, and represent the same set. For sets with many elements, especially those following an implicit pattern, the list of members can be abbreviated using an ellipsis ''. For instance, the set of the first thousand positive integers may be specified in roster notation as Infinite sets in roster notation An infinite set is a set with an endless list of elements. To describe an infinite set in roster notation, an ellipsis is placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of nonnegative integers is and the set of all integers is Semantic definition Another way to define a set is to use a rule to determine what the elements are: Such a definition is called a semantic description. Set-builder notation Set-builder notation specifies a set as a selection from a larger set, determined by a condition on the elements. For example, a set can be defined as follows: In this notation, the vertical bar "|" means "such that", and the description can be interpreted as " is the set of all numbers such that is an integer in the range from 0 to 19 inclusive". Some authors use a colon ":" instead of the vertical bar. Classifying methods of definition Philosophy uses specific terms to classify types of definitions: An intensional definition uses a rule to determine membership. Semantic definitions and definitions using set-builder notation are examples. An extensional definition describes a set by listing all its elements. Such definitions are also called enumerative. An ostensive definition is one that describes a set by giving examples of