kids encyclopedia robot

Naive set theory facts for kids

Kids Encyclopedia Facts

When people first started talking about naive set theory in the 1800s, they used everyday language to describe groups of things, called sets. It was like sorting objects into different collections. They used ideas from discrete mathematics, like Venn diagrams, which are circles that show what's inside a set. This way of thinking about sets was powerful enough for many areas of math and engineering.

What is Naive Set Theory?

Naive set theory is the original, simple way of thinking about sets. It's called "naive" because it doesn't have strict rules or "axioms" about how sets can be formed. It's often compared to axiomatic set theory, which has very strict rules.

The Puzzles of Naive Set Theory

Even though naive set theory was useful, it led to some tricky puzzles, called paradoxes. A paradox is a statement that seems true but leads to a contradiction. Here are a few famous ones:

  • The Burali-Forti paradox, discovered in 1897, showed that you can't create a set of all ordinal numbers (numbers used for ordering, like first, second, third).
  • Cantor's paradox, discovered in 1899, showed that you can't create a set of all cardinal numbers (numbers used for counting, like 1, 2, 3).
  • Perhaps the most famous is Russell's paradox, discovered in 1902. It asks: "What about the set of all sets that do not contain themselves?" If this set contains itself, it shouldn't. If it doesn't contain itself, then it should! This shows a big problem with forming sets too freely.

These puzzles showed that mathematicians needed clearer rules for how sets could be defined.

Trying to Fix the Problems

Mathematicians realized they needed a better way to define sets to avoid these paradoxes.

Dedekind's Ideas

Richard Dedekind (1831-1916) tried to fix these problems in 1888. He suggested using "axioms," which are basic rules or truths, to describe sets. However, his ideas still led to new puzzles:

  • The Richard's paradox, from 1905, involved defining numbers using words, which led to a contradiction.
  • The Berry paradox, from 1908, was about "the smallest number that cannot be described in fewer than twenty English words." This also led to a puzzle.

Zermelo and Fraenkel's Solution

In 1908, Ernst Zermelo published a new set theory that used very strict axioms. These rules limited how sets could be made, helping to avoid the earlier paradoxes. Later, with Abraham Fraenkel, he developed Zermelo–Fraenkel set theory. This became the main way mathematicians thought about sets for most of the 20th century. Even with these new rules, some very complex puzzles remained, as shown by Gödel's incompleteness theorems.

When you learn about sets in school, you might still use some ideas from naive set theory and the definitions given by Georg Cantor, who was a very important mathematician in the early study of sets.

Images for kids

See also

Kids robot.svg In Spanish: Teoría informal de conjuntos para niños

kids search engine
Naive set theory Facts for Kids. Kiddle Encyclopedia.