So we have two infinite parallel lines that are of the same length. We pick one of them and remove every odd inch from it. Then we fill in the gaps that we created using remaining inches. By doing this, the gaps disappear leaving the two lines looking perfectly identical.
The problem is that there isn’t enough inches remaining to fill in the gaps without creating new gaps elsewhere. This illusion is created by moving the gaps out of our sight.
If you don’t see it, it’s not there.
And if you keep pushing things out of your sight, you can keep reassuring yourself they don’t exist.
Especially if this process is an infinite one (:
Here’s the line we started with:
( \bullet \bullet \bullet \bullet \bullet \bullet \cdots )
Now here’s the line with odd inches removed from it:
( \circ \bullet \circ \bullet \circ \bullet \cdots )
There’s an infinite number of inches out of our sight. We don’t see them, we merely see the ellipsis “…” which tells us there is more to this line than what we see. What we want to do now is take three inches from the remaining inches that we don’t see, so that we can fill in the gaps. We can do that, because there’s still an infinite number of inches remaining, so we do that and we get:
( \bullet \bullet \bullet \bullet \bullet \bullet \cdots )
Voila! The line looks like the original one! They now appear to be identical! But what happened to those gaps? Where did they go? Well, they went out of our sight. They didn’t magically vanish. We don’t know exactly where they went, but they are somewhere out of our sight.
So the lines aren’t really identical. They merely look like they are.
The gaps can’t magically vanish. The only thing we can do is push them out of our sight forever thereby creating an illusion that the two lines are identical.
They are not.
This “paradox” is known as Hilbert’s paradox of the Grand Hotel:
en.wikipedia.org/wiki/Hilbert%2 … rand_Hotel
I think Carleas mentioned it somewhere at the beginning of this thread (40-50 pages ago . . .)