There are actually multiple types of infinity. It's totally an abstract concept but it does play a part in math. Here's an example. There are an infinite number of integers. There are also an infinite number of real numbers, which include all integers and all fractions. For two infinities to be "equivalent" you have to show that there is a one-to-one mapping between them. It can be shown that you can't create such a mapping between the integers and the real numbers. No matter how you try, you will always be able to generate a real number (usually by just adding another digit after the decimal point) that isn't mapped to an integer. In fact, you can generate an infinite number of real numbers for each integer. Thus, there are "more" real numbers than integers even though both are infinities. Integers are an order 1 infinity, real numbers are order 2.
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Man, the stuff I remember from college....