Naive set theory is any of several set theories used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and suffices for the everyday use of set theory concepts in contemporary mathematics. Sets are of great importance in mathematics; in modern formal treatments, most mathematical objects (numbers, relations, functions, etc.) are defined in terms of sets. Naive set theory suffices for many purposes, while also serving as a stepping stone towards more formal treatments.
Method A naive theory in the sense of "naive set theory" is a non-formalized theory, that is, a theory that uses natural language to describe sets and operations on sets. Such theory treats sets as platonic absolute objects. 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 – 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 and developed as a formal but inconsistent system by Gottlob Frege in his Grundgesetze der Arithmetik. Naive set theory may refer to several very distinct notions. It may refer to
Informal presentation of an axiomatic set theory, e.g. as in Naive Set Theory by Paul Halmos. Early or later versions of Georg Cantor's theory and other informal systems. Decidedly inconsistent theories (whether axiomatic or not), such as a theory of Gottlob Frege that yielded Russell's paradox, and theories of Giuseppe Peano and Richard Dedekind.
Paradoxes The assumption that any property may be used to form a set, without restriction, leads to paradoxes. One common example is Russell's paradox: there is no set consisting of "all sets that do not contain themselves". Thus consistent systems of (either naive or formal) set theory must include some limitations on the principles which can be used to form sets.
Cantor's theory Some believe that Georg Cantor's set theory was not actually implicated in the set-theoretic paradoxes (see Frápolli 1991). One difficulty in determining this with certainty is that Cantor did not provide an axiomatization of his system. By 1899, Cantor was aware of some of the paradoxes following from unrestricted interpretation of his theory, for instance Cantor's paradox and the Burali-Forti paradox, and did not believe that they discredited his theory. Cantor's paradox can actually be derived from the above (false) assumption—that any property P(x) may be used to form a set—using for P(x) "x is a cardinal number". Frege explicitly axiomatized a theory in which a 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, not necessarily a theory Cantor—who, as mentioned, was aware of several paradoxes—presumably had in mind.
Axiomatic theories Axiomatic set theory was developed in response to these early attempts to understand sets, with the goal of determining precisely what operations were allowed and when.
Consistency 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 is possible to state all the axioms explicitly, as in the case of Halmos' Naive Set Theory, which is actually an 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 does not deal with consistency or completeness of the axiom system. Likewise, an axiomatic set theory is not necessarily consistent: not necessarily free of paradoxes. It follows from Gödel's incompleteness theorems that a sufficiently complicated first-order logic system (which includes most common axiomatic set theories) cannot be proved consistent from within the theory itself – unless it is actually inconsistent. However, the common axiomatic systems are generally believed to be consistent; by their axioms they do exclude some paradoxes, like Russell's paradox. Based on Gödel's theorem, it is just not known – and never can be – if there are no paradoxes at all in these theories or in any sufficiently complicated first-order set theory, again, unless such theories are actually inconsistent. It should be mentioned, however, that results in proof theoretical ordinal analysis are sometimes interpreted as consistency proofs. The term naive set theory is still today 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.
Utility The choice between an axiomatic approach and other approaches is largely a matter of convenience. In everyday mathematics the best choice may be informal use of axiomatic set theory. References to particular axioms typically then occur only when demanded by tradition, e.g. the axiom of choice is often mentioned when used. Likewise, formal proofs occur only when warranted by exceptional circumstances. This informal usage of axiomatic set theory can have (depending on notation) precisely the appearance of naive set theory as outlined below. It is considerably easier to read and write (in the formulation of most statements, proofs, and lines of discussion) and is less error-prone than a strictly formal approach.
Sets, membership and equality In naive set theory, a set is described as a well-defined collection of objects. These objects are called the elements or members of the set. Objects can be anything: numbers, people, other sets, etc. For instance, 4 is a member of the set of all even integers. Clearly, the set of even numbers is infinitely large; there is no requirement that a set be finite.
The definition of sets goes back to Georg Cantor:
Unter einer ‚Menge‘ verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objecten m unsrer Anschauung oder unseres Denkens (welche die ‚Elemente‘ von M genannt werden) zu einem Ganzen.
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