Foundations of Mathematics

Set Theory

Are there different types of infinity in mathematics?

At the end of the nineteenth century, German mathematician George (Georg) Ferdinand Ludwig Philipp Cantor (1845–1918) showed that different orders of infinity existed and that the infinity of points on a line was of a greater order than that of prime numbers. Since that time, mathematicians have managed to divide the topic of infinity into even more precise terms.

To most of us, the universe represents infinity, but in mathematics, infinity is the unbounded quantity that is greater than every real number. In fact, it is called potential infinity in mathematics, in which the potential for infinity exists with natural numbers because you can always mention a number greater than any given number. Directed infinity applies to an infinity in direction z and is an infinite numerical quantity that is a positive real multiple of the complex number z; a directed infinity with an unknown direction is known as a complex infinity. Still another “type” of infinity in mathematics is completed infinity, which refers to the size of an infinite set (such as all the points on a line).


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