Requirements
In the sense of this article, a naive theory is a non-formalized theory, that is, a theory that uses a natural language to describe sets. The words and, or, if ... then, not, for some, for every are not subject to rigorous definition. It is useful to study sets naively at an early stage of mathematics in order to develop facility for working with them. Furthermore, a firm grasp of set theoretical concepts from a naive standpoint is important as a first stage in understanding the motivation for the formal axioms of set theory.
This article develops a naive theory. Sets are defined informally and a few of their properties are investigated. Links in this article to specific axioms of set theory describe some of the relationships between the informal discussion here and the formal axiomatization of set theory, but no attempt is made to justify every statement on such a basis. 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.
As it turned out, assuming that one can perform any operation on sets without restriction leads to paradoxes such as Russell's paradox and Berry's paradox. Some believe that Georg Cantor's set theory was not actually implicated by these paradoxes (see Frápolli 1991); one difficulty in determining this with certainty is that Cantor did not provide an axiomatization of his system. It is undisputed that, by 1900, Cantor was aware of some of the paradoxes and did not believe that they discredited his theory. Gottlob Frege did explicitly axiomatize a theory in which the formalized version of naive set theory can be interpreted, and it is this formal theory which Bertrand Russell actually addressed when he presented his paradox.
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).
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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