Why letting any property define a set leads straight into contradiction
Try forming the set of all sets that are not members of themselves, and you hit Russell's paradox: it can neither contain itself nor fail to. That single puzzle showed that the informal, anything-goes approach to sets pioneered in the late 19th century needed rules about which collections are allowed.
Naive set theory describes sets in plain language rather than formal logic. It covers the familiar tools of discrete mathematics such as Venn diagrams and the algebra of combining sets, and it is enough for everyday mathematical work. Since most mathematical objects, from numbers to functions, are ultimately built from sets, this informal version also serves as a stepping stone to more rigorous treatments, and its notation dominates even in advanced research.
Georg Cantor created the first set theory at the end of the 19th century while studying infinite sets. He defined a set as a gathering into a whole of definite, distinct objects of our perception or thought, called its elements. A set can hold numbers, people or other sets, and need not be finite; the even integers form an infinite set with 4 among its members. Gottlob Frege later turned the ideas into a formal system in his Grundgesetze der Arithmetik, which turned out to be inconsistent.
The trouble lies in assuming every property yields a set. Bertrand Russell aimed his paradox at Frege's formal system, not necessarily at whatever Cantor intended, and some scholars, following Frápolli in 1991, argue Cantor's theory was never implicated. By 1899 Cantor was already aware of two contradictions, the Burali-Forti paradox and one now named after him, yet he saw no threat to his work in them. The first can be produced by taking the property of being a cardinal number as a set.
Axiomatic set theory arose to specify exactly which operations are permitted. The line between naive and formal is blurry: Paul Halmos's book Naive Set Theory is really an informal account of standard Zermelo–Fraenkel set theory, naive only in its everyday language and its silence about consistency. Formal systems carry no guarantee either, because Gödel's incompleteness theorems mean a sufficiently strong system cannot prove its own consistency unless it is inconsistent. The usual axioms do block Russell's paradox and are widely believed consistent.
Source: Naive set theory
