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What are the many types of infinity in Mathematics?

I have heard that, in mathematics, there are many types of infinity. I'm aware of some examples. But I'd like to collect more. Could people please share different types of mathematical infinity?I'm not sure, if you're familiar with Numberphile on YouTube, but they actually made a video about infinity covering different types of infinities:

http://www.youtube.com/watch?v=elvOZm0d4H0Like Stefan, I'm no expert on it, but I'm familiar with a few concepts.

First off, division by zero is defined as "undefined" by mathematicians, but if you take the calculus concept of the limit and take the denominator from some non-zero value closer and closer to zero, you move the fraction's value toward infinity.

The notion that division by zero is "undefined" is curious, though. I certainly like thinking of 1/0 as infinity. But then, if we follow the "rules" of mathematics strictly, then we run into a curious paradox. For instance, n/n = 1 (usually), but what happens if n=0? Then, 0/0 should be "1," but zero in the numerator usually makes the value equal to zero. And then zero in the denominator approaches (arrives at?) infinity. So, 0/0 has three values or none of them -- 0, 1, infinity. And perhaps this gives us an interesting peek at a relationship that may underlie all of reality.

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