Some time ago I had a strange conversation with my son Lucian, who was then nine years old.
“Dad, what is infinity plus infinity? Lucien asked.
“Infinite,” I replied stoically.
But how can a number plus itself be itself? Lucien insisted. “I thought only zero could do that, like in 0+0=0.”
“Well,” I said, “infinity isn’t really a number. It’s more of an idea.”
Lucian rolled his eyes. “So infinity plus one is also infinity?”
Before exploring infinity in Nature, here is a little prelude on infinities in mathematics.
Mathematicians often refer to countable and uncountable infinities. (Yes, there are different kinds of infinities.) For example, the set of all integers (…,-3, -2, -1, 0, 1, 2, 3,…) is a countably infinite set . Another example is the set of rational numbers – numbers of the form p/q constructed from fractions of integers, such as 1/2, 3/4, and 7/8, and excluding division by zero.
The number of objects in each of these sets (also known as the cardinality of the set) is called aleph-0. Aleph is the first letter of the Hebrew alphabet, and it has the cabalistic interpretation of connecting heaven and earth: ℵ. Aleph-0 is infinite, but it is not the greatest possible infinity. The set of real numbers, which includes the sets of rational and irrational numbers (those numbers that cannot be represented as fractions of integers, including √2, π, e, etc.), has cardinality aleph- 1. Aleph-1 is known as the continuum. It is greater than aleph-0 and can be obtained by multiplying aleph-0 by itself aleph-0 times: 1=00.
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Georg Cantor, the pioneering German mathematician who invented set theory, described the continuum hypothesis, which postulates that there is no set with a cardinality between aleph-0 and aleph-1. However, the current results imply that the continuum hypothesis is undecidable – it is neither provable nor unprovable. The human mind is befuddled by ideas of different infinities, even in the formal rigidity of abstract mathematics.
What is the shape of the universe?
What about space? Is space infinite? Does the universe expand into infinity in all directions, or does it fold in on itself like the surface of a balloon? Can we ever know the shape of space?
The fact that we only receive information about what is inside our cosmic horizon, which is defined by how far light has traveled since the big bang, seriously limits what we can know about what is beyond its edge. When cosmologists say the universe is flat, what they really mean is that the part of the universe we measure is flat – or nearly so in the accuracy of the data. We cannot, due to the flatness of our patch, make conclusive statements about what lies beyond the cosmic horizon.
If the universe has an overall shape, could we determine that, stuck as we are in a flat cosmic horizon? If our universe is shaped like a three-dimensional sphere, we may be out of luck. Judging by current data, the curvature of the sphere would be so slight that it would be difficult to measure any indication of it.
An interesting but far-fetched possibility is that the universe has a complicated shape – what geometrists call a non-trivial topology. Topology is the branch of geometry that studies how spaces can continually warp into each other. Continuous means without cutting, like stretching and bending a sheet of rubber. (These transformations are known as homeomorphisms.) For example, a ball without holes can be deformed into a soccer ball-shaped ellipsoid, a cube, or a pear. But it cannot be deformed into a bagel, because a bagel has a hole.
Measure universal signatures
Different cosmic topologies can leave imprinted signatures in things we can measure. For example, if the topology is not simply connected (remember our bagel, which has a hole in its shape), light from distant objects can produce patterns in the microwave background. To use a specific example, if the universe is shaped like a bagel and its radius is small relative to the horizon, light from distant galaxies may have had time to wrap around multiple times, creating multiple images identical as reflections in parallel mirrors. In principle, we could see such ghostly images or patterns in a mirror, and these would provide information about the overall shape of space. So far, we have not found any such indicator.
Since we do not see such images, can we conclude that space is flat? You can never measure anything with absolute precision, so we can never be certain, even if the current data strongly points to zero spatial curvature in our cosmic horizon. In the absence of positive curvature detection, the question of the shape of the space is therefore unanswered in practice. Is it something unknowable? It seems like. Something quite radical would have to step in to make it known, like a theory that can calculate the shape of space from first principles. So far, we don’t have such a theory. Even if one day one happens, we will have to validate it. This gives us all sorts of problems, as we discussed recently.
The conclusion may be disappointing, but it is also extraordinary. The universe may be spatially infinite, but we cannot know that. Infinity remains more of an idea than something that exists in physical reality.