Intuitions about infinity
The idea of infinity is inextricable from the concept of countability.
In other words: there is no notion of the sublime, the cosmos, etc. without picturing some poor sucker trying to make his way up the endless hill, to count the rice on the chessboard, etc.
The infinite is not a standalone quality; it is a failure mode of perception.
There are as many kinds of infinities as there are failure modes we can imagine, plus the one failure mode we can’t imagine.
I am on an island with as many beaches as there are kinds of infinity.
Ailana Beach is quiet, perfectly still. I pick up a handful of yellow sand and begin to count the grains. By the end of the day I’ve counted two handfuls of sand. I scan the beach and it’s hopeless – there are so many more handfuls of sand than there are days remaining in my life. I’ll never get there.
On Brick Beach I’ve set up a state of the art counting machine that counts the grains incredibly fast. But every wave deposits a new batch of sand, and the sand is accumulating at a rate exponentially greater than the the machine’s counting speed. I’ll never catch up.
Carraway beach is famous for its jet black sand. Each grain has a sleek, almost alien sheen to it. As I pick one up to count it, I realize that this grain is itself a cluster of micro-grains packed so tightly together that there are at least as many micro-grains in a single grain as there are grains of sand on the beach. I put one of the micro-grains under a microscope and see a packed cluster of nano-grains. I’ll never get anywhere.
Duquesne Beach is a tiny white beach. As I approach it my heart leaps; it seems that there are only a few grains of sand on the beach. But as soon as I sit down to count, something strange happens. Every time I look at a grain of sand, more grains pop out of it. Merely trying to count the sand causes it to multiply. There’s nothing I can do.
And so on.