Those of you who read this blog (all 1 of you!) know that the number of integers is infinite and the number of irrational numbers is infinite, but the latter infinity is greater than the former infinity. I tend to think of this as mathematical shenanigans. The idea that one infinity is larger than another just doesn't meet the "beauty" test that accompanies mathematics. In the past, I've argued that infinities don't exist, so we can make contradicting remarks about them to our hearts content. It's similar to asking the question "Can God make a rock so big even he can't lift it?" The words make sense and are in the right order, but the semantics of the sentence are off. They produce a barber paradox that belongs to the realm of meaninglessness. They show that Goedel is alive and well and relevant for today's meta-mathematical problems, if only we'd heed his words. Ok, back to infinities.
First, can we create a more intuitive reason for why one infinity might be greater than the other, a reason that doesn't fall back on establishing a one-to-one correspondence with the integers?
First, let's establish that the integers exist in one-dimensional space. In fact, they exist on a number line which is the definition of one-dimensional space :-) But, let's consider a slightly different definition of dimensionality. In this definition, we'll look at the number of infinite dimensions. In the case of integers, the length of the integer is finite. No matter how big the integer becomes, there is always a finite number of digits. Even if the integer explodes to a googolplex digits, there is still a finite number of them. Therefore, the only infinity is in how many integers there are, the size of the actual integer is finite. So, there is only one dimension of infinity.
Let's now look at rational numbers. In the case of rational numbers you might say that there are two dimensions of infinities. The first dimension is the number of rational numbers. There are an infinite number of rational numbers. The second dimension is that some rational numbers have an infinite decimal expansion. For example, 1/3 has an infinite expansion of 0.333333333... Therefore, rational numbers have two dimensions of infinity and should be larger than integral numbers, right? Not so fast. That's just their decimal expansion. If you keep them in their functional form, then we get a different story. All rational numbers can be expressed as the division of two integers. Moreover, we know that both integers have a finite number of digits. Finally, we know that adding two integers with a finite number of digits will produce a third integer with a finite number of digits. Therefore, there is a representation that is finite in length for every rational number. That leads to the logical conclusion that all rational numbers have one dimension of infinity.
Next up, irrational numbers. Irrational numbers extend to positive and negative infinity, giving one dimension of infinity. In addition, they have an infinite expansion in every base, which gives them a second dimension of infinity. So, we can easily see why there are more irrational numbers than rational numbers, because we allow irrational numbers to have an infinite expansion!
So, if we can't represent irrational numbers with a fixed number of rational numbers, what can we represent them with? Why, an infinite number of rational numbers, of course! For example, PI can be represented by the following series (one of many): PI = 4 * (SUM[k=0 to inf] (-1^k)/(2k+1))
So, in essence, we have an infinite set of numbers each composed of an infinite set of numbers. Two dimensions of infinity!
But wait! If we can create a series to represent each irrational number, does that mean that there exists a representation that is not infinite and therefore we can count them, similar to the rational numbers above? No. With rational numbers there was a finite number of integers that created the rational number. With irrational numbers there is an infinite number of rational numbers.
So, it is true that the number of irrational numbers exceeds the number of rational numbers, right? Well, maybe. I think it is fair to say that an infinitely expanded irrational number doesn't exist, so we're back to the land of Goedel. It is worth thinking of an irrational number as a function of an infinite number of rational numbers much the same as a rational number is a function (division) of two integers. Functions are often useful in mathematical manipulations, but that doesn't make them "real". The number 3 is a physical, real number. You can count out 3 things. The number PI is not a physical, real number. We can only approximate it. In fact, complex analysis is based around the number i, which is a function (sqrt) applied to -1. As long as we don't expand the function, we can do mathematics with it, but if we ever need to expand it our equations blow up. The same thing is true with irrational numbers, they are mathematical niceties. Abstractions that we can manipulate as long as we don't look too closely. Similarly, questions about abstractions such as if one abstraction is infinitely larger than another abstraction requires you to look to closely, so you get crazy results, just like if you had really taken the sqrt of -1 or divided by infinity (another function).
So, the next time you see PI, take it for what it is, a function that can be evaluated to the necessary precision. A mathematical abstraction that can be admired from afar. Just don't get too close.
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