Objects generally fall into three classes. There are those objects that obey the laws of the set, and there are those objects which do not obey these laws. A set cannot contain objects that do not obey the rules of set. Therefore, no set can contain objects which are not set-the objects which are the members of the “infinity set” or the “hyperreal numbers.”
Infinity and hyperreal numbers are the three main classes of objects that do not obey the rules of the set, as they are not governed by the laws of set. These objects are called the “finite and transcendental” classes of objects, which means that they are not governed by the rules of set and, therefore, cannot be contained by a set.
Infinity is the number of sets which can be made up out of the one set. If a set of objects is made up of infinitely many objects, this set is said to be infinitesimal. Therefore, if an object belongs to an infinite number of sets, it is said to belong to an infinity. Therefore, a set consisting of an infinity is called a “Infinity Set.”
In addition to being the largest set that can exist, finite sets are the smallest set that can exist. A finite set consists of one object and no more. A set can only have one member and no more members. If there were infinitely many objects, a set would have infinitely many members and no members.
It is possible for a finite set to include a non-finite object. In other words, while two objects can belong to the same set, a set can contain other objects apart from themselves. This type of set is called a triplicate set, a collection of three objects joined together by one member.
Finite sets are also classified according to their size. Sets of objects of the same size can either be infinitesimals or polyinfinitesimals, while sets that have smaller sizes can either be larger or smaller.
Sets are the foundation of all mathematics. All objects in mathematics are classified under sets. Sets theory, and therefore, all knowledge about mathematics, is based on sets.
A set can be thought of as a collection of objects, and so the objects in the collection can be thought of as “sets.” The size of a set is equal to the largest number of objects which belong to that set. So if an object belongs to a particular set, the size of the set is also the largest number of objects that belong to that set.
Sets are also referred to as “classes” because they consist of objects and not individuals. If there are no persons in the set, then the set is called a class, and is not an individual. Therefore, the set of all students at Harvard University is not an individual, but a class.
Sets can be used to define relationships among objects. A relationship between a set and its objects is called a set relation, while a relationship between two sets is referred to as a set relationship.
For example, if an apple belongs to a set, then the set of all apples is a class relationship between a pair of oranges. Similarly, if two sets are related by a class relationship, then they are said to belong to the same class.
Sets are used to classify all types of objects in nature. Examples of objects that can be classified into sets are sets of all objects and their properties, sets which have a certain size, sets of objects and their properties, and sets of properties. For instance, the numbers of the natural numbers and the properties of the real numbers can be classified into sets.