Infinity a Number?

I'm sure you've been told before not to treat infinity as a number. So, what exactly would happen if we thought of infinity as number?

First off, let us set ∞ as a number, with the special properties that ∞ is equal to ∞ + 1. (For proof on this, go to Hilbert Hotel Paradox.)

First, we can prove that ∞ = ∞ + C, where C is any given integer.
- Because ∞ = ∞ + 1, then ∞ = (∞ + 1) + 1, and ∞ = ((∞ + 1) + 1) + 1, and so on
- You end up with ∞ = ∞ + 1 + 1 + 1 + ... + 1, where 1 is written C times.
- And because ∞ is a number, we can write ∞ = ∞ + ∞

Next, because ∞ is a number, then obviously ∞ ÷ ∞ = 1

However, we can also prove that ∞ ÷ ∞ = 2
- ∞ ÷ ∞ = 2
- ∞ = 2∞
- ∞ = ∞ + ∞, which we proved to be true already

Then, we can prove that ∞ = 0
- ∞ ÷ ∞ = 2
- ∞ = 2∞
- 0 = ∞ (subtracting ∞ from each side)

From this, we can easily prove that ∞ = 1
- ∞ = 0
- ∞ + 1 = 0 + 1
- ∞ = 1

Finally, because 0 = ∞, and 1 = ∞, then 0 = 1

Of course, we could have gotten this from the beginning equation: ∞ = ∞ + 1, by subtracting ∞ from both sides

So, when we make infinity a number, we get all sorts of weird proofs, like that 0 equals 1, which is obviously not true. Below shows a few of the different problems we can get into.

So in conclusion, unfortunately, we can not treat infinity as a number.

On the next page, I will introduce bijections, to compare different infinities.