Naive VS Axiomatic set theory

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Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. [1]
Naive set theory is defined informally, in natural language.

Unlike axiomatic set theories, which are defined using formal logic.
Axiomatic set theory was developed in response to early attempts to understand sets,
with the goal of determining precisely what operations were allowed and when.

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A naive theory in the sense of "naive set theory" is a non-formalized theory, that is,
a theory that uses a natural language to describe sets and operations on sets.

The words and, or, if ... then, not, for some, for every are treated as in ordinary mathematics.
As a matter of convenience, use of naive set theory and its formalism prevails even in higher mathematics
โ€“ including in more formal settings of set theory itself.

The first development of set theory was a naive set theory.
It was created at the end of the 19th century by Georg Cantor as part of his study of infinite sets[3]
and developed by Gottlob Frege in his Grundgesetze der Arithmetik.