Infinity is a concept that is hard to grasp. But it is also fun in sort of that brain teaser kind of way. Is it a number? Is one infinity greater than another infinity?
Steven Strogatz tells a story in his book the Joy of X about responding to a letter from a mom on behalf of her 6 years old son asking if infinity is odd or even. He responded by saying that if infinity were odd then two times infinity would be even, which is not possible since they are both infinity. The mom said her son was happy with that explanations since it kind of meant infinity was big enough to be both odd and even, which was not Strogatz's intent.
It is important to remember that infinity is not an actual number. To me personally, I consider a number to be something that is finite. Something you can represent with numerical notation. 1.7498 is a number to me, just as much as 7/189 and 5 are numbers. Infinity doesn't play by the same rules as numbers do. For example it is neither even nor odd. If infinity were a number by my definition, we would be able to count to it, or at least order it, which we obviously can not. Cantor, looked at infinity through the eyes of set theory. He defined cardinality as how many elements were in a set, and he viewed infinity as a measurement of how many elements were in certain sets. If we have the infinite set of all integers and then the infinite set of integers that are divisible by 100, which set will be larger? There is a much larger part of my mind that wants to say the set of all integers will be 99 times bigger obviously! But infinity is not a number, we can't compare it that way. They are both infinity. Neither set is larger than the other. The set of all integers contains numbers that are closer together on the number line than the set which are divisible by 100, but we can steadily add numbers to each set at the same rate for an infinite amount of time. We can have corresponding items in each set such as, 1 corresponds to 100, 2 corresponds to 200, and so forth. I'm not trying to tell you, if we counted each number in each set they would have the same amount of elements in each set. That is not the point, first of all we can't count to infinity. Therefore we can't compare infinities, and therefore both sets are infinitely large. They just happen to be different infinities. I like to think of infinity as a sort of all or nothing concept. Something is either infinite or it is finite. Two different infinite sets are both infinite, they are both the 'all' they just happen to be different 'alls'.
Content: you are entitled to your opinions, sir, even if they are not shared by the followers of Cantor. But you could include some of Cantor's ideas for comparison. Eg.if numbers are cardinality of sets, infinity is a number. If that is true, then there are different infinite numbers.
ReplyDeleteConsolidation: I think if you got at what your idea of a number is, that would tie up the ideas you're discussing.