Page "Naive set theory" Paragraph 5

Axiomatic set theory was developed in response to these early attempts to study set theory, with the goal of determining precisely what operations were allowed and when.

Today, when mathematicians talk about " set theory " as a field, they usually mean axiomatic set theory.

Informal applications of set theory in other fields are sometimes referred to as applications of " naive set theory ", but usually are understood to be justifiable in terms of an axiomatic system ( normally the Zermelo – Fraenkel set theory ).