Naive set theory as a first order theory

Question

In the Wikipedia article about Russell's paradox the authors present the naive set theory as a first order theory (as far as I understand), but without references. Can anybody share some references where the naive set theory is presented that way (for explaining Russell's paradox) so that we could input them into the Russian Wikipedia article?

Answer 1

There is no standard definition of "naive set theory". I am going to make this a community wiki answer in which people can record various definitions.

Some definitions of "naive set theory" include:

• Informal systems of set theory that are not stated in any formal logic. In such system the comprehension refers informally to "properties" but is not stated as an axiom scheme. Depending on the author, these may have a restricted comprehension axiom, and thus be intended to be consistent, or may have an unrestricted comprehension axiom, and thus be intended to be inconsistent.
• Formal systems of set theory with unrestricted comprehension (which may thus be inconsistent).
• Halmos' book Naive Set Theory has its own sense of "naive" which is quoted below.

Halmos' Naive Set Theory

The following quotes are from the preface of Halmos' Naive Set Theory.

The purpose of this book is to tell the beginning student of advanced mathematics the basic set-theoretic facts of life, and to do so with the minimum of philosophical discourse and logical formalism. (p. v)

 

In set theory "naive" and "axiomatic" are contrasting words. The present treatment might best be described as axiomatic set theory from the naive point of view. It is axiomatic in that some axioms for set theory are stated and used as the basis of all subsequent proofs. It is naive in that the language and notation are those of ordinary informal (but formalizable) mathematics. A more important way in which the naive point of view predominates is that set theory is regarded as a body of facts, of which the axioms are a brief and convenient; in the orthodox axiomatic view the logical relations among various axioms are the central object of study. Analogously, a study of geometry might be regarded as naive if it proceeded on the paper-folding kind of intuition along; the other extreme, the purely axiomatic one, is the one in which axioms for the various non-Euclidean geometries are studied with the same amount of attention as Euclid's. The analogue of the point of view of this book is the study of just one sane set of axioms with the intention of describing Euclidean geometry only. (pp. v-vi).

Halmos briefly sketches the syntax of first-order logic in order to state the "Axiom of Specification", so it could be argued that his axiom is intended to be a scheme in first-order logic.