What would you see if you came to the edge of the Universe? It’s hard to imagine so it’s tempting to conclude that the Universe doesn’t have an edge and therefore that it must be infinite. That’s not a necessary conclusion however. There are things that are finite in extent but still don’t have an edge, the prime example being the surface of a sphere. It’s got a finite area but when you walk around on it you’ll never fall over an edge. The question of whether the Universe is finite or infinite is one that still hasn’t been answered, and there are mathematical models that allow for both possibilities. More generally, the question of whether any infinite quantities can arise in the Universe is a deep one. In April this year philosophers, cosmologists and physicists came together at the University of Cambridge, as part of a conference series on the philosophy of cosmology, in order to discuss it. Plus went along to find out more (and you can also listen to the interviews we did in our podcast).
People have been studying infinity and its relation to reality for a long time. “The idea of studying infinities in physics really began with Aristotle,” says the Cambridge cosmologist John D. Barrow. “Aristotle made a clear distinction between two types of infinity. One he called potential infinities and he was quite happy to allow for those to appear in descriptions of the world. These are just like lists that never end. The ordinary numbers are an example; one, two, three, four, five, and so on, the list goes on forever. It’s infinite, but you never reach or experience the infinity. In a subject like cosmology, there are lots of infinities like that and most people are quite happy with them. For example, the Universe might have infinite size; it might have an infinite past age, it might be destined to have an infinite future age. These are all potential infinities, so they don’t bite you as it were, they’re just ways of saying that things are limitless, they’re unbounded, like that list of numbers.”
While most people are happy to accept that potential infinities may exist, we still don’t know whether they actually do. “When you look at the Universe, how far you can see is strictly limited, because the Universe has been in existence for a finite time, for around 14 billion years,” says George Ellis, a cosmologist from the University of Cape Town. “Light travels at the speed of light, so you can only see out to a distance essentially of 14 billion light years; it’s a little bit bigger but basically that’s it. There’s no way you can see to infinity. It’s like [looking out] from a tower on the surface of the Earth; you can see to the horizon, but you can’t see beyond. In that case you can get on a plane and fly to the other side. In the case of the Universe, the scale is such that we can’t move; we’re stuck at one point and we can only see the Universe from one point out to a finite distance.”
Infinity (symbol: ∞) is an abstract concept describing something without any limit and is relevant in a number of fields, predominantly mathematics and physics. The English word infinity derives from Latininfinitas, which can be translated as “unboundedness”, itself calqued from the Greek word apeiros, meaning “endless”.
In mathematics, “infinity” is often treated as if it were a number (i.e., it counts or measures things: “an infinite number of terms”) but it is not the same sort of number as the real numbers. In number systems incorporating infinitesimals, the reciprocal of an infinitesimal is an infinite number, i.e., a number greater than any real number. Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities). For example, the set of integers is countably infinite, while the infinite set of real numbers is uncountable. [Wikipedia]
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