Vitali set

A Vitali set is a special group of numbers in mathematics. It cannot be measured using a common tool called Lebesgue measure. This idea was found by a mathematician named Giuseppe Vitali in 1905.

These sets have more numbers than we can count. There are actually an infinite number of Vitali sets. To show that these sets exist, mathematicians use a rule called the axiom of choice. The sets are strange and hard to describe.

Measurable sets

Some groups of numbers have a clear idea of "length" or "size." For example, the group of numbers from 0 to 1 has a length of 1. If we have two separate groups, like numbers from 0 to 1 and from 2 to 3, we can add their lengths together to find the total length.

But some special groups are harder to measure. For example, the numbers we can write as fractions between 0 and 1 are very common. But when we use a special way to measure them called Lebesgue measure, their total length is 0. Sets that fit well with this measuring method are called "measurable." However, not all sets behave this way, and this leads to tricky questions in mathematics.

Construction and proof

A Vitali set is a special group of numbers between 0 and 1. It is built in a way that for every real number, there is exactly one number in the set.

Vitali sets are interesting because they cannot be measured using a common way to measure lengths in math. This shows that some sets in math are very unusual.

Properties

A Vitali set does not have the property of Baire. We can also show that each Vitali set has a Banach measure of 0. This is okay because Banach measures only add up for some numbers, not for all of them.

Role of the axiom of choice

To build a Vitali set, mathematicians use something called the axiom of choice. This helps us see if some sets can’t be measured in a special way. The answer is yes, if we think very large numbers, called inaccessible cardinals, fit with our main rules of set theory, known as ZFC.

In 1964, a mathematician named Robert Solovay made a special version of set theory without the axiom of choice. In this version, all sets of real numbers could be measured. For this, he thought inaccessible cardinals do not cause problems with the rules of set theory. Most experts think this idea is right, but it can’t be proven using ZFC alone. Later, in 1980, another mathematician named Saharon Shelah showed that Solovay’s result depends on this idea about inaccessible cardinals.