Ordinal numbers have two related meanings. Colloquially, an ordinal number is a number that tells the position of something in a list, such as first, second, third, etc. This basic understanding extends to the meaning of ordinal numbers in set theory. In an ordered set, that is a collection of objects placed in some order, ordinal numbers (also called ordinals) are the labels for the positions of those ordered objects.
For example, the natural numbers have a standard ordering, which is expressed by the sequence of labels . The integers double both as elements of and as labels for the order on . This makes sense all the way up to infinite numbers. Ordinals are defined by the ordinals that come before. For instance, the ordinal can be identified as the set .
The set of natural numbers is a countably infinite set. The "size" of the natural numbers as a countably infinite set is a common standard to categorize 2 types of infinite sizes: countable and uncountable. If elements of another set can be put into one-to-one correspondence with the natural numbers, that set also has a size of countably infinite.
The position of elements in finite sets are fairly obvious: they start with the natural numbers , meaning and so on to Note can be an ordinal, too. After the natural numbers is the first infinite ordinal , and then , etc. all the way to or . Then , etc. all the way to . Note, it's important to express this as or not or Ordinals are not commutative, and would be treated the same as , whereas is an ordinal number decrementing one more than .
This is beginning to form a pattern where and are natural numbers. This collection of ordinals is itself a set assigned the ordinal . There is also . And this can be continued indefinitely far. In fact, ordinals are designed to be continued indefinitely for there always to be another ordinal after the .
Ordinals are linked to a unique kind of set called a well-ordered set. A well-ordered set is actually a type of totally-ordered set. A total-ordered set essentially means that given two elements, one will be smaller and one larger and that this distinction can be defined in a coherent manner. Formally, let be a set. A total order on is a binary relation, denoted by , such that for all , the following axioms hold:
- Either or .
- If and , then .
- If and , then .
For example, the standard ordering on is a total order . As a consequence, the dictionary order, meaning the commonly understood order like on is also a total order. This is defined by setting iff in the standard ordering on for all .
Back to well-ordered sets and ordinals! A well-ordered set is a total-ordered set with no infinite decreasing sequences. Ordinals may label the elements of the well-ordered set to measure the length of the whole set by the least ordinal that is not a label. Note that with the standard ordering is not well-ordered, since for every element , the element satisfies . However, the natural numbers with their natural ordering are well-ordered, by the well-ordering principle.
Two sets and , with total orders and respectively, are called order-isomorphic if there exists a bijection such that implies for all .
Suppose that a sequence of ordinal labels has been constructed. Then, every ordinal describes a well-ordered set, namely the set of all ordinals less than . This set, which we also denote by , can be thought of as the canonical example of a particular well-ordered set, in the sense that it represents the equivalence class of all well-ordered sets that are order-isomorphic to it.
To illustrate this, consider the finite ordinal , which one identifies with the ordered set of ordinals less than it, namely . Then, any well-ordered four-element set (0 is the ordinal label for a non-element, for the zeroth element) is order-isomorphic to ; for example, the four-element set with well-order permits the order-isomorphism given by , , , and .
Based on this intuition, one formally defines ordinals as equivalence classes of well-ordered sets. If and are well-ordered sets, then one writes if and are order-isomorphic; this relation is an equivalence relation, and the equivalence classes obtained are called ordinals. Denote by the equivalence class of , i.e. the ordinal containing the well-ordered set .
In this formalism, the finite ordinals are precisely . The smallest infinite ordinal is . For an example of an infinite ordinal larger than , consider the set with well order Since does not have the same cardinality as , the ordinal does not equal . Usually, one denotes .
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