Ordinal theory is a branch of mathematics, and it is an intuitive concept to study order with binary relations (such as greater than and better than). It provides a formal framework for describing statements such as "This is less than" or "This is before". This paper introduces the research field and gives the basic definition. A list of ordered theory terms can be found in the Ordered Theory Glossary.

In mathematics and related fields, such as computer science, the concept of order is everywhere. The first order often discussed is the standard order of natural numbers learned in primary school, such as "2 is greater than 3", "10 is greater than 5", or "Do I have more cookies than you?" . This intuitive concept can be extended to the order of other groups of numbers, such as integers and real numbers. The idea of greater than or less than another number is one of the basic intuitions of the number system (compared with the counting system) (although you are usually interested in the actual difference between two numbers, which is not given by the order). Other familiar sorting examples are the alphabetical order in the dictionary and the generation of a family.

The concept of ordinal number is very general, which goes beyond the content with direct and intuitive sense of order or relative quantity. In other cases, ordinal numbers may capture the concept of inclusion or specialization. In the abstract, this kind of order is equal to the subset relationship, such as "pediatrician is doctor" and "circle is only an ellipse in special cases".

Some sequences, such as "less than" and alphabetical order of natural words, have a special property: each element can be compared with any other element, that is, it is smaller (earlier) and more consistent. For example, consider the order of subsets on a set: although both bird sets and dog sets are subsets of animal sets, neither bird nor dog constitutes a subset of the other. Ordinal numbers similar to the "subset" relationship of elements are called partial orders; The comparable ordinal number of each pair of elements is the complete ordinal number.

Ordinal theory captures the intuition of ordinal numbers generated from these examples under general conditions. This is achieved by specifying that relationships (such as "≤") must be attributes of mathematical ordinal numbers. This more abstract method makes sense, because people can deduce many theorems under general settings without paying attention to any specific order details. These insights can be easily transferred to many less abstract applications.

Driven by the wide practical application of ordinal numbers, many special types of ordinal numbers have been defined, some of which have become their own mathematical fields. In addition, ordinal number theory is not limited to various ranking relationships, but considers appropriate functions between them. A simple example of the ordinal theoretical attribute of a function comes from the analysis that monotone functions are often found.