Mind games about infinity, eternity, and sustainability.

There are many things that most Mormons seem to believe even though the scriptural evidence is less than solid. One of these is that there is an eternal exponential increase in the number of people in existence. People don’t usually state it like this; instead they usually talk about our Heavenly Father raising use to be heavenly parents like himself, meaning that we will eventually raise children who will also become parents, and so on ad infinitum.

This brings to mind a set of paradoxes and puzzles about infinity. “‍If life expands exponentially without end, is that sustainable?‍” Clearly, if the universe is finite the answer to that is “‍no.‍” But I personally have never found any argument that the universe is finite to be even slightly convincing. Let us therefore look at this question in terms of an infinite universe, but a fixed infinite universe, one in which no new stuff to make people out of is being or can ever be created.

There are two cases worth a look. In the first, the amount of life present today is finite. In the other, there is already an infinite supply of life.

If the supply of life is finite right now, it will remain finite no matter how long it multiplies and will thus never fill an infinite universe at any finite time in the future.

If the supply of life is infinite right now, we can further ask “‍what percentage of the space or matter in the universe is used by life?‍” This could be zero (the way that the integers are 0% of the real numbers) or a finite number. If zero, we are still fine for any arbitrarily-long finite time: no matter how many times we double in number, we’ll still be only 0% of the universe. But if it is not zero, we might be in more trouble.

Let’s suppose that 1 part in a million is used by life right now. Let’s also suppose that the amount of life doubles every century, such that in 29 centuries half of the person material will be used up. In 30 centuries, will we run out of stuff to make people with?

Of course not. We’ll just ask everyone to help us out in the following way. We’ll have every person line with a person-sized lump of unused stuff between each one. Then we’ll ask everyone to pass one lump down toward the end of the line. The first person will have two lumps, and everyone else will still have one. We’ll excuse the first person and ask everyone to do the same thing again. The next person will now have two lumps, and we repeat for the third, and fourth, and so on. When we are done, we’ll have gone from one half used to only one third used. Anytime the population gets to big we’ll just repeat the process.

But maybe you don’t like the fact that this passing of matter will take infinitely long. That’s fine, we can avoid that too. Let’s number everyone, from 1 to ∞, and also number each lump of matter. We’ll tell each person n to grab lumps 2n and 2n + 1. They’ll all do this at the same time and Presto! Twice as much matter.

We can do it without the numbering too. Have everyone grab a lump so that all the lumps are taken. Have everyone face the same direction with their lump in front of them and two people behind them. This will be a branching pattern, like a tree, except that being infinite it will keep branching forever. Now have everyone turn around and take the two lumps from the two people behind them. Since it’s an infinite tree, there are two people behind each of them so they all immediately get two lumps. They can then put these two on their other side and repeat to get four, or eight, or however many they need.

Of course, this also works the other way. Do any of these in the other order (matching two people to a lump) and poof! half the unused stuff is gone.

Do these arguments work? Is it true that an infinite queue of people can only have one end? Or is it meaningful to talk about an infinite like of people with someone on each end, or an infinite tree with one root and an infinite number of leaves infinitely far away?

The simple answer: no one knows.