# Definition:Naive Set Theory

## Contents

## Definition

**Naïve set theory**, in contrast with axiomatic set theory, is an approach to set theory which assumes the existence of a universal set, despite the fact that such an assumption leads to paradoxes.

A popular alternative (and inaccurate) definition describes this as a

*non-formalized definition of set theory which describes sets and the relations between them using natural language.*

However, the discipline is founded upon quite as rigid a set of axioms, namely, those of propositional and predicate logic.

## Also see

- Results about
**naïve set theory**can be found here.

## Linguistic Note

The pronunciation of **naïve** is in two syllables, approximately **naa- eeve**, and means

**simple**in the sense of

**unsophisticated**.

Try not to pronounce it **nigh- yeeve**.

In natural language it is usually used in the sense of **lacking worldly wisdom**.

The correct rendition of the word **naïve** is with a (diacritic) diaeresis on the **i**, that is: **ï**.

This has the effect of indicating that the pair of letters involved need to be pronounced as separate syllables. Spelt **naive**, the word would be expected to be pronounced something more like **nave**.

$\mathsf{Pr} \infty \mathsf{fWiki}$ usually follows what appears to be standard practice, and renders the word with the normal unadorned **i**, but on occasion the correct form can be seen.

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 2$: Introductory remarks on sets: $\text{(g)}$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**naive set theory** - 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (next): $\S 1$: Naive Set Theory - 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*... (next): Appendix $\text{A}.1$: Sets - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (next): $1$: Set Theory and Logic