Elementary Theory of the Category of Sets (ETCS)

Elementary Theory of the Category of Sets (ETCS) is one way to define Set Theory which uses only sets and morphisms in its definition. ¹

To define Set Theory using only sets and morphisms, you can use the following statements: ¹

• An element in a set is defined as a function from a singleton set to that element.
• A singleton set is defined as every set having exactly one unique function that goes to the singleton set.
• An empty set is defined as a set that has a function from it to any other set.

The singleton set and empty set are also known as the terminal object and the initial object. To do function application, define a function from the singleton set to the element in a set that we want to select. Then, that can be combined with a function. ¹

Under this theory, the definitions of a singleton set and an empty set can be considered to be duals of each other (See: Category Theory). ¹