My mind is a swirling maelstrom of facts, useless information, questions, opinions, and dreams. There is one question that seems to pop out every now and then, and every time I manage to find the answer, I become so afraid of it that the question goes into remission for a while until I decide to look up the answer again. It’s the kind of question that kind of makes you think about the meta-magnitude of mathematics.
For those of you who aren’t necessarily mathematically-minded, let me give you a refresher.
We all know about numbers, but remember that all the numbers are divided into sets that fit into one another.
One such set is the set of integers. Integers are whole numbers that stretch from positive infinity to negative infinity; think of the number line, and picture the middle of it as 0. It extends eternally to the left and right, with negative integers on the left and positive integers on the right.
The set of integers is part of the set of real numbers, which contain rational (numbers that can be represented as a fraction, or ratio) and irrational numbers (that can’t be expressed as a ratio, such as the popular mathematical constant π).
Back to the question I was talking about earlier. Clear your mind, and really think about this one, because I’ll give you one answer I received to it, and I want to know whether or not you agree and why. Either way, I find it interesting. The question is:
Which set is larger… the set of all integers, or the set of all real numbers between 0 and 1?
I submit that the answer is the latter.
Is your mind blown yet?
No? Let me help it along a little.
Let’s consider, first, the size of the set of all integers. As I said before, it stretches from negative infinity to positive infinity, and contains whole numbers. It can’t be quantified, but if you had an infinite amount of time, you could count them, even if you started at an immensely large x and stopped at –x.
However, what about between 0 and 1? At first, one would say, “Nothing. When counting, you start with 0, and go to 1.” But remember, I said the set of all real numbers, and that includes fractions and irrational numbers. So to keep it simple, I can give you two numbers between 0 and 1; let’s go with 0.25 and 0.26. Not even realizing the fact that if you stay at that number of significant digits, you still have a wide berth between 0 and 1, but there are numbers between those… 0.251 and 0.252. The set of integers doesn’t have this problem. With integers, if you have x, you have nothing between it and x + 1.
Hopefully you see where I’m going with that. No matter which two decimals you compare, you can always go down one further layer of granularity and get a real number between the two. So not only do you have an infinity between 0 and 1, but you have an infinity between every set of decimals you can possibly postulate. Therefore, the set of real numbers between 0 and 1 is a much larger set.
A bit of research has led me to something called Aleph Numbers, and how set sizes are described with cardinality.
Even if you aren’t good at math, go down the rabbit hole for a little while, and see how infinitely deep it is.