Some time ago I had a strange conversation with my son Lucian, then nine years old.
“Dad, what is infinity plus infinity?” Luciano asked.
“Infinite,” I replied stoically.
“But how can a number plus itself be itself?” Luciano insisted. “I thought only zero could do this, as in 0 + 0 = 0.”
“Well,” I said, “infinity isn’t really a number. It’s more of an idea.”
Luciano rolled his eyes. “So is infinity plus one also infinity?”
Before exploring infinity in Nature, here is a prelude to infinity in mathematics.
Mathematicians often refer to countable and uncountable infinities. (Yes, there are different types of infinity.) For example, the set of all integers (…,-3, -2, -1, 0, 1, 2, 3,…) is an infinite countable 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 cardinal of the set) is called aleph-0. Aleph is the first letter of the Hebrew alphabet, and has the kabbalistic interpretation of connecting heaven and earth: ℵ. Aleph-0 is infinity, but not the greatest possible infinity. The set of real numbers, which includes the sets of rational and irrational numbers (the numbers that cannot be represented as fractions of integers, including √2, π, and so on), has a cardinal of 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 posits that there is no set with a cardinal between aleph-0 and aleph-1. However, the current results imply that the continuum hypothesis is undecidable – it is neither demonstrable nor improbable. The human mind is confused with ideas of different infinities, even within the formal rigidity of abstract mathematics.
What is the shape of the universe?
And what about space? Is space infinite? Does the universe stretch toward infinity in all directions, or does it fold in on itself like the surface of a balloon? Can we know the shape of space?
The fact that we only receive information from what lies within our cosmic horizon, which is defined by the distance light has traveled since the big bang, seriously limits what we can know about what lies beyond its edge. When cosmologists say that the universe is flat, what they really mean is that the portion of the universe we measure is flat — or close to that within the accuracy of the data. We cannot, from the flatness of our smear, make any conclusive statements about what lies beyond the cosmic horizon.
If the universe is globally shaped, could we determine that, trapped as we are within a flat cosmic horizon? If our universe is shaped like a three-dimensional sphere, we might be out of luck. Judging by the current data, the curvature of the sphere would be so small 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 geometers call a nontrivial topology. Topology is the branch of geometry that studies how spaces can continually deform into one another. Continuously means without cutting, like when you stretch and bend a sheet of rubber. (These transformations are known as homeomorphisms.) For example, a ball with no holes can be deformed into an ellipsoid shaped like a soccer ball, a cube, or a pear. But it cannot be deformed into a bagel, because a bagel has a hole.
Measuring universal signatures
Different cosmic topologies can leave signatures imprinted on 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 bagel-shaped and its radius is small compared to the horizon, light from distant galaxies may have had time to curl up a few times, creating several identical images, like reflections from parallel mirrors. . In principle, we could see these mirror images or ghostly patterns, and this would provide information about the overall shape of the space. So far, we have not found such an indicator.
Since we don’t see such images, can we conclude that space is flat? We can never measure anything with absolute precision, so we can never be sure, even if current data strongly point to zero spatial curvature within our cosmic horizon. In the absence of positive curvature detection, the question of the shape of space is therefore unanswered in practice. Is it something unknowable? It seems to be. Something quite drastic would need to step in to make it known, like a theory that can calculate the shape of space from first principles. So far, we have no such theory. Even if one day it arrives, we will need to validate it. This presents us with 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. Infinity remains more of an idea than something that exists in physical reality.