On an intuitive level, density/countability of Q

[johnqwertyful]

Something I've thought about for a bit on an intuitive level is about the countability/density of Q. It seems almost contradictory. I know what countable means, I know what dense means. I know how to prove Q is dense. I know how to prove Q is countable. But on an intuitive level, it's hard to wrap my mind around it.

Intuitively, if something is countable, given any element in the set and enough time, you can count until you reach it. Like for N, given ANY natural number, no matter how big, and infinite time, and I could count until I reach it.

But with Q, let's say I wanted to count to 1. There's an infinite number of elements between 0 and 1, so I would be there forever, never being able to reach 1? What about 1/2? I shouldn't be able to reach there.

The way to reconcile it in my head is that the bijection with N isn't order preserving, right? In the way you count Q on [0,1], you would superficially go through it, then again, and again and again. It's still a little unclear though. Anyone able to reconcile them intuitively?

 

[johnqwertyful]

After three quarters of undergrad analysis, going on to grad analysis next quarter, I feel like I should know this lol.

 

[deluks917]

You are correct. There is no order preserving bijection from N to Q. Why is the density of Q combined with countability confusing you? You did not explain this.

One issue many students I know learnign analysis have said confuses them is the following:

Consider the Ball of size [itex]\frac{1}{2^n}[\itex]. The union of all such balls have length at most 1 (this is made rigourous with measure theory but it works as you would guess). However since all any real is arbitarily close to a rational. So doesn't this say the length of the real line is less than 1?

Do you see what is wrogn with their understanding of density.

 

[lurflurf]

You are pretty close. We start counting the rationals (pick your favorite bijection). We reach any particular rational in a finite number of steps. We do not (obviously) count any infinite set (such as the rationals in (0,1)) in a finite number of steps. This is not unlike the integers. We reach any integer n a finite number of steps, but we cannot reach all even numbers in a finite number of steps. This seems to be more about countability than density, but think of several examples, not just Q. I like m/2^n and x^(m/n) among others.

 

[blue_raver22]

I completely understand your confusion and difficulty in wrapping your mind around the countability and density of Q. These concepts can be quite abstract and difficult to grasp intuitively. However, through mathematical proofs and logical reasoning, we can understand and accept these concepts as true.

To start, let's define what countability and density mean in the context of Q. Countability refers to the ability to assign a unique natural number to each element in the set. In other words, if we can create a one-to-one correspondence between the elements in Q and the natural numbers, then Q is considered countable. Density, on the other hand, refers to the closeness of elements in a set. If a set is dense, it means that between any two elements in the set, there exists an infinite number of elements. In simpler terms, there are no "gaps" in the set.

Now, let's consider Q. We know that Q is countable because we can create a one-to-one correspondence between the elements in Q and the natural numbers. This is known as the Cantor's Diagonal Argument. However, this does not mean that we can easily count to any element in Q. This is where density comes into play. Yes, there are an infinite number of elements between 0 and 1 in Q, but we can still count to any specific element by using a rational number line. This means that even though there are an infinite number of elements between 0 and 1, we can still reach any specific element by counting in a systematic way.

To further clarify, let's use your example of counting to 1 in Q. Yes, there are an infinite number of elements between 0 and 1, but we can still reach 1 by systematically counting using smaller and smaller fractions. For example, we can start by counting 1/2, then 1/4, then 1/8, and so on. This process may take an infinite amount of time, but we can still reach 1. Similarly, we can count to any other element in Q by using a similar systematic approach.

In terms of the bijection with N not being order preserving, this is correct. This means that the order of the elements in Q does not necessarily correspond to the order of the natural numbers. However, this does not impact the countability or density of Q.

In summary, the countability and density of Q can be reconciled