Infinity is a number, oh wait, no its not. Or is it? This is one of the many debates still causing mathematicians problems. What really is infinity? To me, infinity is an idea. It is a something that we can never actually reach but we pretend that it is there.
I recently watched a lecture by Manil Suri that blew my mind. He talked about different paradoxes that involve infinity. The first one was the Hotel Infinity. If every room in the Hotel Infinity is booked and somebody comes in and asks for a room. The receptionist says "Here's your room key" and gives the man a key. How is this possible if the hotel is booked? Well the answer is surprisingly simple. Instead of making the man walk down the hallway to his room, they make everybody in every room move down one. This way the first room in the hotel is vacant for the man and the man won't die making it up to his room.
Another concept the Suri talked about that we covered in class is that when we look at the set of the natural numbers versus the set of the even natural numbers, which one is bigger? They each go to infinity right? The more obvious choice would be to choose the set of natural numbers since there are twice as many as just the even natural numbers. In fact, we are able to match the set of even natural numbers in one to one correspondence with the natural numbers such that 2 matches up with 1, 4 matches up with 2, 6 matches up with 3 and so one. In this case, they both go to the same infinity and neither set is larger! So that leads to the question, is there any set of numbers that go larger than infinity? The answer is yes. The set of real numbers from 0 to 1 is larger than infinity. This is because there are so many real numbers between 0 and 1 that it is impossible to set them up in a one to one correspondence with any other set. This then causes the cardinality of this set to be larger than any other cardinality that we can find.
The idea of infinity was credited to Georg Cantor. Cantor's work was mostly in set theory. This is how the thoughts of infinity started to come about. He started to ask questions about continuity and infinity. Working with one-to-one corresponding sets made him start to think that they could go on forever. This idea did not sit very well with most people around his time.
Infinity is such a crazy number or idea. It is hard to wrap my head around something so large that once you do, infinity is actually bigger than that.