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Are there more integers or more even integers?
Seems like a simple question, right? After all, every even integer is an integer but what about all the even integers? So there are more integers than there are even integers, right? But wait a second. How many even integers are there? An infinite number. And how many integers are there? An infinite number. Hmmmm....
"Infinity," says math student A, "is just a term... there's no way you can actually show me that there is the same number of each."
"Okay, let's play..." says math student B. "Give me an integer, and I'll give you an even integer that corresponds to it. And if two of your integers are different, I guarantee that my two even integers will be different."
Math Student A: Okay... 1
Math Student B: 2
A: I'm beginning to see what you mean. But let's consider some of the set theory we learned in math class. The set of even integers is contained in the set of integers, but is not equal to that set. So the two sets can't be the same size.
(Who's right? What kind of sets did the teacher put on the board in class? How do these sets differ from those?)
The paradox characterized by the above problem puzzled mathematicians for centuries. At its core lay that troubling concept that haunts all of mathematics: infinity. In 1874 Georg Cantor worked out a system of degrees of infinitythat solved the problem once and for all and greatly increased mathematicians' understanding of infinity and set theory.
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