A mind-blowing fact about infinity

What can you say about infinity? Well, the word has a pretty lengthy Wikipedia page, so clearly the answer is "a lot."

Which makes sense, since if there's one thing we know about infinity, it's this: infinity is freaking big.

Or at least, limitless. I suppose something could be infinitely small -- getting smaller and smaller without limit. But this would mean that it's a freaking big bit of small.

God supposedly is infinite.

Infinitely loving, infinitely knowing, infinitely powerful. Heck, God probably has an infinite number of positive qualities, being so infinite. This is assuming that God exists.

We also have to assume infinity exists.

Reading the "A Window on Infinity" chapter in physicist David Deutsch's The Beginning of Infinity book, I learned that there's some disagreement about whether infinity is really real, or just conceptually real.

There is a philosophy of mathematics called finitism, the doctrine that only finite abstract entities exist. So, for instance, there are infinitely many natural numbers, but finitists insist that that is just a manner of speaking.

They say that the literal truth is only that there is a finite rule for generating each natural number (or more precisely, each numeral) from the previous one, and nothing literally infinite is involved.

I don't feel qualified to comment on the dispute between finitists and infinitists. The whole notion of infinity blows my mind when I ponder it. So it's tough for me to say anything coherent about a word -- infinity -- which seems utterly divorced from what I experience in everyday life.

Limits. Discrete numbers of things. Boundaries. Horizons.

Still...

I love th! e notion of infinity. I want the finitists to be wrong. I get a thrill up my psyche's spine when I try to envision something that isn't really a some, nor a thing, having no limit.

I haven't finished reading Deutsch's "A Window on Infinity" chapter. But here's my favorite mind-blowing passage so far.

In mathematics, infinity is studied via infinite sets (meaning sets with infinitely many numbers). The defining property of an infinite set is that some part of it has as many elements as the whole thing.

Whoa. Wow.

Every part of an infinite set is just as infinite'y as the whole thing. As soon as this idea entered my mind, it seemed super significant to me. Problem is, I can't say why. I get a feeling along the lines of "universe in a grain of sand."

However, with infinity seemingly there's no boundary that could define a universe, or a grain. Seemingly. I'm not mathematically astute enough to know if this is true.

Realizing that most readers of his book would be as math-challenged as I am, Deutsch helpfully recounts a thought experiment by mathematician David Hilbert. Here's a sample of what goes on in the experiment.

He [Hilbert] imagined a hotel with infinitely many rooms: Infinity Hotel. The rooms are numbered with natural numbers, starting with 1 and ending with -- what?

The last room number is not infinity. First of all, there is no last room. The idea that any numbered set of rooms has a highest numbered member is the first intuition from everyday life that we have to drop.

Second, in any finite hotel whose rooms are numbered from 1, there would be a room whose number equalled the total number of rooms, and other rooms whose numbers were close to that: if there were ten rooms, one of them would be room number ten, and there would be a room number nine as well.

But in Infinity Hotel, where the number of rooms is infinity, all the rooms have number! s infini tely far below infinity.

Whoa. Wow.

That last sentence seems really, really, really significant to me. Why, it could be the key to understanding everything. Or, nothing. Which doesn't take anything away from the sentence's significance.

After all, nothing is as close to infinity as everything is. I think. Can't be sure. Time to stop blogging and have a glass of wine. That will bring me closer to understanding what infinity is all about.

Of course, that new understanding distance still will be infinitely far below infinity. So I might as well watch TV while drinking the glass of wine.

Reading a holy book. Sitting at the feet of a holy person. Meditating or praying in a holy fashion. Doing any or all of those things would still leave me infinitely far below infinity.

Remember, in the Infinity Hotel all the rooms have numbers infinitely below infinity.