Cantor, Georg, << KAHN tawr, gay AWRK >> (1845-1918), a German mathematician, created a fundamental branch of mathematics and logic called set theory. Using set theory, Cantor also developed revolutionary ideas related to the concept of infinity (see Infinity ).
A set is a collection of elements. An element can be a physical object, like an apple, or an abstract idea, like a number. A set may be finite or infinite. A finite set has a limited number of elements. Examples of finite sets include the set of the numbers {1, 2, 3} and the set of the letters {a, b, c}. An infinite set has an unlimited number of elements. An example is the set of all even numbers: {2, 4, 6, 8, 10, 12, …}. Such a set goes on forever.
Cantor explored the idea that two sets can have a one-to-one (1:1) correspondence with each other. For example, the set {1, 2, 3} and the set {a, b, c} have a 1:1 correspondence, because each element in one set can be paired with exactly one element from the other. For finite sets, a 1:1 correspondence can only exist if the two sets have the same number of elements.
The principle of 1:1 correspondence can be extended to infinite sets, with surprising results. An example is the 1:1 correspondence between the set of all counting numbers {1, 2, 3, …} and the set of even numbers {2, 4, 6, …}. The correspondence pairs each counting number with its double and each even number with its half. This shows that the two sets have the same number of elements—even though the set of all counting numbers contains the entire set of all even numbers in addition to all the odd numbers.
This seeming paradox—that part of an infinite set can be as big as the whole set—confused mathematicians. Before Cantor, mathematicians had concluded that it simply made no sense to talk about the number of elements in an infinite set, thus avoiding the paradox. Cantor, on the other hand, defined an infinite set as a set that can be matched one-to-one with a part of itself. Embracing the paradox of infinite sets allowed Cantor to revolutionize the way people studied infinity.
Cantor also showed that not all infinite sets are the same “size,” or cardinality. Using the idea of 1:1 correspondence, Cantor showed that infinite sets can have different “degrees” of infinity. For example, the set of real numbers—that is, all the points that form the number line—cannot be matched up 1:1 with the counting numbers. The real numbers are “uncountably infinite.” They thus have a higher cardinality than “countably infinite” sets that can be matched 1:1 with the counting numbers.
Cantor was born on March 3, 1845, in St. Petersburg, Russia. He spent most of his professional life in Halle, Germany. Cantor’s ideas were profound, but they attracted much criticism and controversy. Cantor suffered from bouts of depression. He died in a mental institution on Jan. 6, 1918.
See also Fractal ; Set theory (History) .
