Cantor created a theory about the possible sizes of infinite sets. With the results of his research, he concluded that some infinite sets can be larger than others. Even though this idea sounds absurd, and even though it brought him a lot of ridicule from other mathematicians, it nevertheless helped to pave the way for the mathematics of set theory. "A short summary is that Cantor invented set theory, and then used it to study the construction of finite and infinite sets, and their relationships with numbers. One of the very surprising conclusions was that you can compare the size of infinite sets: two [infinite] sets have the same size if there’s a way to create a one-to-one mapping between their members. An infinite set A is larger than another infinite set B if every possible mapping from members of B to members of A will exclude at least one member of A. Using that idea, Cantor showed that if you try to create a mapping from the integers to the real numbers, for any possible mapping, you can generate a real number that isn’t included in that mapping – and therefore, the set of reals is larger than the set of integers, even though both are infinite."^{2} |
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