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A naive set theory is not necessarily inconsistent, if it correctly specifies the sets allowed to be considered.
This can be done by the means of definitions, which are implicit axioms.
It can be done by systematically making explicit all the axioms, as in the case of the well-known book Naive Set Theory by Paul Halmos, which is actually a somewhat ( not all that ) informal presentation of the usual axiomatic Zermelo – Fraenkel set theory.
It is ' naive ' in that the language and notations are those of ordinary informal mathematics, and in that it doesn't deal with consistency or completeness of the axiom system.
However, the term naive set theory is also used in some literature to refer to the set theories studied by Frege and Cantor, rather than to the informal counterparts of modern axiomatic set theory ; care is required to tell which sense is intended.

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