Examples
The collection of all algebraic objects of a given type will usually be a proper class. Examples include the class of all groups, the class of all vector spaces, and many others. In category theory, a category whose collection of objects forms a proper class (or whose collection of morphisms forms a proper class) is called a large category.
The surreal numbers are a proper class of objects that have the properties of a field.
Within set theory, many collections of sets turn out to be proper classes. Examples include the class of all sets, the class of all ordinal numbers, and the class of all cardinal numbers.
One way to prove that a class is proper is to place it in bijection with the class of all ordinal numbers. This method is used, for example, in the proof that there is no free complete lattice.
Read more about this topic: Class (set Theory)
Famous quotes containing the word examples:
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)