Vitali set

In mathematics, a Vitali set is a special kind of group of numbers that cannot be measured using a common tool called Lebesgue measure. This idea was discovered by a mathematician named Giuseppe Vitali in 1905. The Vitali theorem tells us that such sets do exist, even though they are very unusual.

Each Vitali set contains more numbers than we can count, and there are actually an infinite number of these sets. To show that these sets exist, mathematicians use a basic rule called the axiom of choice. This helps them build the sets, even though the sets themselves are quite 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 just add their lengths together to find the total length.

But what about special groups, like only the numbers we can write as fractions between 0 and 1? These numbers are very common, but when we use a special way to measure them called Lebesgue measure, their total length ends up being 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 that fits a certain pattern with rational numbers.

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 and hard to describe with normal tools.

Properties

A Vitali set does not have the property of Baire. By changing the way we prove this, we can also show that each Vitali set has a Banach measure of 0. This is okay because Banach measures only add up for a few numbers at a time, not for all of them together.

Role of the axiom of choice

To build a Vitali set, mathematicians use something called the axiom of choice. This makes us wonder if we really need this axiom to show that some sets cannot be measured in a special way. The answer is yes, if we believe that very large numbers, called inaccessible cardinals, fit with our main rules of set theory, known as ZFC.

In 1964, a mathematician named Robert Solovay created 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 assumed that inaccessible cardinals do not create any problems with the rules of set theory. Most experts think this assumption is correct, but it cannot be proven using ZFC alone. Later, in 1980, another mathematician named Saharon Shelah showed that Solovay’s result depends on this assumption about inaccessible cardinals.