The set **N**, that is the set of all positive integers, has infinite number of numbers, that is the number of numbers in set **N** is ∞, but none of the numbers in set **N** is ∞. As we remember, ∞ is not a number, especially it’s not an integer in the case of set **N**.

Let us assume that we start from finite number of people, but all of the people are immortal and at finite rate the number of people increases and that there’s endlessly time.

Can we even ask, that what is the number of people in these conditions ”eventually”? There’s no end, but we can’t just say that eventually the number of people would be infinite. Or can we?

*Image courtesy of Stuart Miles at FreeDigitalPhotos.net*

In the natural numbers set, there’s always a greater number, there’s no limit – but as said, none of the numbers is ∞. So would this be the case in the number of people in immortality – it would always get bigger and would never reach an end and never be ∞.

If the number of people would eventually be ∞, when would it be ∞ for ”the first time”? If the number of people would eventually be ∞, there would be a moment when the number of people would have a start but not an end; before that moment the number of people had a start and an end.

According to a joke, when asked when infinity is reached, a mathematician says ”never”, a physicist says ”sometime”, an engineer would say ”very soon”. 🙂

Well, this is just humor, the set theory is not model of reality.

Zeno of Elea (ca. 490 – 420 BC) is known as Greek philosopher, who’s paradoxes have troubled also modern thinkers for a long time.

Zeno aimed at to show contradiction between plurality, motion and infinity with his paradoxes.

Motion is impossible, because whatever moves must reach the middle of its course *before* it reaches the end; but *before* it must have reached the quarter-mark, and so on, *infinitely*. Hence the motion can never even start.

So there are infinite number of sub-courses to go, before the end has been reached.

It was because of lacking concept of (mathematical) limit, these paradoxes saw the daylight. Arbitrary small changes couldn’t be explained. Though, there are views, that even modern calculus doesn’t explain *completely* Zeno’s paradoxes! So is the motion still an illusion? 🙂

*Image courtesy of Stuart Miles at FreeDigitalPhotos.net*

Isaac Newton founded differential and integral calculus in the 17th century.

First, ∞ is *not* number. It’s symbol for infinity. For example in the natural numbers set **N** (the set of all positive integers), there are infinite number of numbers, but none of them is ∞.

But now to the example…

What is 1^{∞}? Layman probably would say that that’s 1, but according to mathematicians 1^{∞} is not defined; we can’t say (at least by our understanding on infinity we have at the moment) what it is.

But what if we get ”close” to 1^{∞}? Let’s examine following:

From arithmetic operations in infinity we remember, that

So what we have at the first limit sentence is ”1^{∞} ” and for

an exact value can in fact be determined! It’s the Napier’s number *e*, that’s an irrational number i.e. its decimal representation is infinite. It’s approximately 2.718281828459.

What I presented here, I came familiar with on my first university year.