Table of Contents
How do you prove some infinities are bigger than others?
If you’re given an infinite set, there is a simple method to make a larger infinity: take its power set, which is always of higher cardinality. So not only some infinities are larger than others, but there is no a “largest” inifinity, you can always create a larger one.
Can some infinities be larger than others?
Infinity is a powerful concept. There are actually many different sizes or levels of infinity; some infinite sets are vastly larger than other infinite sets. The theory of infinite sets was developed in the late nineteenth century by the brilliant mathematician Georg Cantor.
Who proved different sizes of infinity?
mathematician Georg Cantor
As German mathematician Georg Cantor demonstrated in the late 19th century, there exists a variety of infinities—and some are simply larger than others.
Are all infinities the same size?
Cantor showed that there’s a one-to-one correspondence between the elements of each of these infinite sets. Because of this, Cantor concluded that all three sets are the same size. Mathematicians call sets of this size “countable,” because you can assign one counting number to each element in each set.
How many different infinities are there?
Three main types of infinity may be distinguished: the mathematical, the physical, and the metaphysical.
Can you compare two infinities?
So yes, we can compare infinites, but no, the set is not smaller than in this sense. This is counter-intuitive but it is true: an infinite set equals in size one of its subsets. In fact, one can show that the size of every infinite set equals the size of (at least) one of its subsets!
How many years did Georg Cantor think about infinities?
He thought about infinities for more than twenty years, and his obsession with infinities made him spending his last years in mental institutes. Georg Cantor introduced the idea of an infinitely enumerable set. He said that a set whose elements can be paired off with the natural numbers.
How did Cantor prove that there are more numbers than natural numbers?
In fact, Cantor showed, there are more real numbers packed in between zero and one than there are numbers in the entire range of naturals. He did this by contradiction, logically: He assumes that these infinite sets are the same size, then follows a series of logical steps to find a flaw that undermines that assumption.
Do all infinities come in different sizes?
Strange but True: Infinity Comes in Different Sizes. That assumption, however, is not entirely sound. As German mathematician Georg Cantor demonstrated in the late 19th century, there exists a variety of infinities—and some are simply larger than others. Take, for instance, the so-called natural numbers: 1, 2, 3 and so on.
How did Cantor prove bijective correspondence?
However, Cantor showed that both have the same cardinal, and therefore the same infinite number of elements. To demonstrate this, he paired each of the elements that form a set with the elements of the other, which is known as establishing a bijective function (or one-to-one correspondence) between both sets.