Is the Real Number Line Actually Infinite?

[Stoney Pete]

Hi guys,

Here is a little puzzle I have been wondering about. I can't solve it, perhaps you can help.

We know there are infinitely many real numbers on the number line. Indeed, already between 0 and 1 there are infinitely many real numbers. So if a real number is a point on the number line, there must be infinitely many points between 0 and 1. This is only possible if the points have zero extension. Now, infinity times zero is still zero. So the number line must have zero length! This is obviously absurd. But where is the false premiss?

Source: Yanofsky, The Outer Limits of Reason.

 

[mfb]

Now, infinity times zero is still zero

Why? Infinity is not even a real number. How do you define multiplication on a set that both contains infinity and zero?

 

[Mark44]

Hi guys,

Here is a little puzzle I have been wondering about. I can't solve it, perhaps you can help.

We know there are infinitely many real numbers on the number line. Indeed, already between 0 and 1 there are infinitely many real numbers. So if a real number is a point on the number line, there must be infinitely many points between 0 and 1. This is only possible if the points have zero extension. Now, infinity times zero is still zero.

No. You can't use arithmetic with infinite values. Infinity times zero is one of several so-called indeterminate forms, such as $[\infty \cdot 0]$, $[\infty/\infty]$, $[0 / 0]$and $[\infty - \infty]$. They are usually written inside brackets to indicate that they aren't actually numbers.

Indeterminate forms typically show up in limits in calculus

So the number line must have zero length! This is obviously absurd. But where is the false premiss?

Source: Yanofsky, The Outer Limits of Reason.