14 July, 2009
The manager of Hilbert’s Hotel had a problem. He prided himself on the fact that his hotel, having an infinite number of rooms, was always open and had space for new guests. It was what made Hilbert’s that little bit special. The hotel could afford to run since an infinite number of guests had arrived the previous week and generated an infinite amount of income, which was just as well as the cleaning bill was infinite too. The manager was going through the bills when disaster struck – another guest arrived! Flustered for a second, since all the rooms were taken, the manager realised a solution. He moved the guest in room 1 to room 2, the guest in room 2 to room 3 and so on and so on. This left room 1 free for the new guest, and the crisis was averted. When later that day an infinite number of new guests arrived the manager took their appearance in his stride, and knew just how to fit these visitors into his very unique hotel.
The above idea, the Hotel with infinite rooms, was created by the German mathematician David Hilbert to highlight some of the paradoxical thinking that is required when dealing with the infinite. Mathematicians use infinities in many different ways. One of the first ways we are introduced to a mathematical notion of infinity is in calculus. Infinities here are used as limits for sequences. Many sequences will tend towards an actual number, for instance the sequence of 1, followed by a half, then a quarter, then an eighth and so on, will tend towards zero. The sum of such a sequence will tend towards 2, and we can define 2 as the limit of the sum of the series. When a series diverges, say 1 + 2 + 4 + 8 + 16 + 32 … then we can define the limit as infinity. This particular use of infinity doesn’t seem to bring in too many paradoxes, and we can safely use calculus without the spectre of the infinite hotel.
Is space infinite? This is still an open question in cosmology. The evidence at the moment implies that there is no boundary to the universe. This does not however immediately imply that the universe is infinite, it could exist in a particular finite shape. To use an analogy, from a two-dimensional viewpoint there is no boundary on a globe. We can walk any way we wish around the world and we will never leave it, yet it is finite. The latest evidence from the cosmic microwave background radiation suggests however that the universe is flat, and this leaves open the possibility that it is infinite in size. Not that we will be able to see much of this infinite universe, beyond what is called the ‘observable universe’. Due to the action of dark energy the universe is expanding at such a rate that eventually the only stars we will be able to see will be those of our local galaxy group. Our observable universe is shrinking in size within this infinite universe, although it will take trillions of years until all we see is the one galaxy we are in.
Speaking of trillions of years, is there an infinite amount of time? It doesn’t seem too difficult to imagine that there will be no end to time, things will just keep happening. But what about the other way? Can we imagine no beginning to time? That one seems a little bit more fraught with difficulty, though for psychological reasons rather than any physical reasons. The evidence does indicate that there was a beginning to time as we understand it. As for an end, when the universe experiences heat death and there is no more energy flowing around, when even protons have decayed into nothingness, will there be anything occurring to distinguish one moment from another? Will time have any meaning then?
Theologians often use the idea of infinity. In the Bhagavad Gita, Arjuna is shown a vision of the ultimate – Brahman, that which is without end, that which is without qualities. When approaching this infinite being we cannot say anything about what it is, it would be a paradox to do so. All that can be said is what is is not, Neti, Neti, not this, not this. Well of course it doesn’t take too long for people to start saying what the ultimate is, and it happened fairly quickly in some forms of Hinduism. Meanwhile the monotheist philosphers of the west had decided that God was three things – all good, all knowing and all powerful. Omnibenevolent, omniscient and omnipotent. By giving God these infinite characteristics the theologians could start to answer some of the difficult question. Infinities lead to Hilbert’s Hotel and paradoxes, and what better way to describe God than through paradoxes. How does the theist explain suffering in a world created by an all good God? One imaginative way is to say that we offend God in some small way, but due to his infinite nature such an offence warrants an infinite punishment. His all good nature is what is stopping such an awful punishment but leaves in the suffering we see in the world. Once infinities are posited, explanations can get quite paradoxical. And would a God who wasn’t all powerful, say 99% powerful, be worthy of worship? Is an unending nature required for a God?
The infinite is a concept that has flowed through our culture since ancient times, from Zeno’s paradoxes to Escher’s paintings. It is useful, indeed necessary for vast amounts of our scientific understanding of the world. And yet, one can still posit that the universe has no actual infinities, that they are a trick of mathematics, and the actual is finite indeed. Like this article, there may be an end to everything.