Infinity is a truly beautiful concept in the mathematical sciences. And yet, despite this fact, we still have plenty of things that we just don’t know, or understand about infinity.
Let me give an example of how difficult it is to rap your head around infinity. A question that pops up frequently is the question, “Are the numbers 1 and 0.99999 … (with the number 9 repeating to infinity) equal?” I posed this question at one time to some intelligent students who want to be scientists and engineers and nearly half of them got it wrong.
The correct answer is yes the two numbers are equal, but people hesitate because of a lack of understanding of infinity. The argument I always heard against the two numbers being equal is that well once you stop you would have to round the number up, but there is where the reasoning goes wrong. The number goes on forever, there is no stopping.
I may as well offer a “proof” that the two numbers are equal for those who are interested. Think about the number 1/3 which we know is the decimal 0.3333… with the number 3 repeating to infinity. Now if I multiply 1/3 by the number 3 we know that the answer is 1. But if you multiply it in decimal form you get 0.9999… so the two numbers must be equal. (This is not a formal mathematical proof but it illustrates the point).
The confusion runs deeper, even as we get into more advanced mathematics. For instance we also can discover that there are actually two different kinds of infinities, ones that we label as countable infinities and uncountable infinities. Let me do my best to introduce these things, imagine the natural numbers that is the numbers 1, 2, 3, 4, 5 … and so on to infinity. The natural numbers are what we call countably infinite (the intuition is that you can “count” these numbers as any school child could). Now let’s imagine the set of integer numbers which can be thought of like …-3, -2, -1, 0, 1, 2, 3 … so you can go backwards to negative infinity or forwards to positive infinity. It would seem as though the integers have twice as many numbers as the natural numbers, except this is wrong, they actually have the same number of elements in the set (I strongly encourage you to Google Hilbert’s Hotel which is a great example of how this “paradox” can be resolved).
Now the set of all real numbers falls into the other category, this one is uncountably infinite. You see the rational numbers are countably infinite, but you can fill in numbers even between rational numbers such as the number pi and the number e, and the square root of 2. Because you fill in these numbers it becomes essentially impossible to count them (hence the name uncountable).
Now that your head is probably spinning (I know mine does every time I try to think about infinity) let’s throw in an idea that Georg Cantor came up with. What if there are infinities in between these two infinities? You know, infinities that are bigger than countably infinite and uncountably infinite in size. This is an idea that has been kicking around for a long time, that there could be a spectrum of infinities (or an infinite number of different sizes of infinite sets).
Now even though Cantor came up with the idea, he didn’t believe it was true, he felt that there were indeed only two sizes to infinity but he couldn’t prove it. Recently a couple of mathematicians proved just that, though there is still some skepticism in the mathematical community to go around, it seems, almost surprisingly that there are only two sizes to infinity.
Perhaps the next logical question is why? But in any case there are still plenty of mysteries to infinity and also plenty of misconceptions about it to go around.