Who was Georg Cantor?

Georg Cantor was a German mathematician who founded, in 1874, a new branch of mathematics called Set Theory. He used this Set Theory primarily as a tool for discovering many interesting theorems about infinity.

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Before Cantor, everyone understood "infinite" as simply meaning "not finite" and that's all there was to it. For example, the set of all integers, or the set of all even numbers, or the set of all fractions, or the set of all real numbers--are all infinite sets. No one questioned whether these four examples of infinity were all of the same size, or whether we had different sizes of infinity.

Cantor did a lot of thinking about these examples and others as well. He discovered that the elements of certain sets, such as the set of all integers, the set of all even numbers, or the set of all fractions, could be arranged into an infinite list. However, the real numbers cannot be arranged into an infinite list; no matter how you try to build such a list, there will always be some real numbers that will be left out.

The method most often used to prove that the real numbers cannot be put into an infinite list is called diagonalization.

Sets whose elements can be arranged into a list are called countable or enumerable sets. (Many mathematicians don't like the word countable and would prefer a less confusing word, like listable; nevertheless, the word "countable" has been around for a very long time and probably won't be replaced anytime soon.) Sets whose elements cannot be arranged into a list (such as the set of real numbers) are called uncountable or non-enumerable sets.

So, what we have now is a truly remarkable discovery! There is not just one infinity! There are different levels of infinity! There is the countable infinity, and there is the uncountable infinity, and the uncountable infinity is bigger than the countable infinity!

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Georg Cantor introduced the idea of Cardinality as a way to express the size of a set. For finite sets, the cardinality of a set is simply the number of elements in the set.

For infinite sets, cardinality becomes a bit more complicated. The simplest example of an infinite set is the set of Natural Numbers = {1, 2, 3, 4, ...}. Cantor called the cardinality of this set ℵ₀. (ℵ is the Hebrew letter Aleph.)

If we let ℕ represent the set of Natural Numbers, then we can form the set ℘(ℕ) (which we call the Power Set of ℕ, and it's defined to be the set of all subsets of ℕ.) Cantor showed that this set necessarily is of a higher cardinality than ℵ₀, and called the cardinality of this set ℵ₁.

In fact, if we happen to have a set of cardinality ℵ_n, we can always create its Power Set (i.e., the set of all subsets of that set); it will necessarily have to have a cardinality that is higher, and we'll call that cardinality ℵ_n+1.

Thus we see that we have different levels of infinity, with cardinalities ℵ₀, ℵ₁, ℵ₂, ℵ₃, ℵ₄, ...

What Georg Cantor was able to do here is truly remarkable! He showed that there are an infinite number of levels of infinity!

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Another very remarkable discovery of Georg Cantor was a subset of the unit interval [0,1] that does not contain any subintervals (in fact, it's of measure zero, or has length zero) and is uncountable. This came to be famously known as the Cantor Set.

Using the idea of the Cantor Set, a very interesting continuous function can be constructed from the unit interval [0,1] to the unit interval [0,1]. Its graph is a continuous curve connecting the points (0,0) to (1,1), but is flat almost everywhere. This graph is called the Devil's Staircase.