Is space infinite?

Georg Cantor

According to Georg Cantor, one of the first to study the concept of infinity in depth, there is not just one concept of infinity, but three different ones. Let's see how he expresses it:

The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent other-worldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number, or order type. I wish to make a sharp contrast between the Absolute and what I call the Transfinite, that is, the actual infinities of the last two sorts, which are clearly limited, subject to further increase, and thus related to the finite. (Georg Cantor, Gesammelte Abhandlungen, Springer, 1980. Translation taken from Rudy Rucker, Infinity and the Mind, Princeton University Press, 2004).

It’s clear, right? On the one hand there is God, who is Infinite and Absolute. On the other, the infinites of physics. On the third place, those of mathematics. Cantor wants to mark the difference between God and the infinites associated with creation, whether in concrete form (those of physics, if there are any) or in abstract form (those that we can imagine).

The question: Do infinites exist? could therefore be broken down into three different questions: a) Does God exist? b) Are there infinities in the world around us, i.e. in physics? c) Are there infinities in mathematics?

I have talked about question a) in several posts in this blog. In this series of posts I am going to talk about question b). We will talk about question c) in later posts.

In relation to the possible infinities in physics, the question is again subdivided into three: a) Is space infinite? b) Is time infinite? c) Can density be infinite?

In relation to this topic, it is worth mentioning Aristotle:

It is incumbent on the person who specializes in physics to discuss the infinite. And to inquire whether there is such a thing or not, and, if there is, what it is. (Physics III, 202b34).

In this post I’ll consider the first question, which in turn can be divided into two: a) Is the extension of space infinite? b) Is space infinitely divisible? The first is the subject of cosmology. The second of quantum physics. We’ll speak here about cosmology.

George Ellis

The possible infinity of space may lead to surprising consequences, such as the possibility that what is happening here and now is also happening on infinite planets in an infinite universe. Regarding this, there are two main theories: one, proposed by George Ellis and Geoff Brundrit, which is based on the theory of general relativity; the other, by Joan Garriga and Alexander Vilenkin, based on quantum mechanics. Against both theories, Francisco José Soler Gil and I published an article, whose reference appears at the end of this text, to which the following paragraphs belong:

Along the history of physics, once and again situations have emerged where infinities seemed impossible to avoid. Most of them have been abandoned and new improved theories have avoided the infinite… The experience provided by the history of physics seems to indicate that [general relativity and quantum theory] could be simple approximations to a third more general theory, which would unify them and eliminate the infinities. Perhaps, when this theory is found, the cosmologic infinities may also disappear… although of course, not all physicists share this attitude of total rejection of the existence of infinities in nature.

It is important to realize that, even if we start from the standard cosmological model, the possibility that the universe is infinite can only be deduced if an additional postulate is accepted, namely the so-called ‘cosmological principle’, which asserts that the universe is homogeneous and isotropic at sufficiently large scales… However, its introduction gives rise to serious problems. In fact, the cosmological principle is being increasingly questioned from both sides: theoretical arguments and observational evidence.

The current standard cosmological model leaves open the possibility that the universe may be infinite. This would be compatible with a universe with null or negative curvature (open universe), while a universe with positive curvature would imply a finite but unbound space (closed universe) which starting at some point would start to shrink and would end in a Big Crunch. The curvature of the universe is currently thought to be approximately zero, but that word (approximately) leaves open the possibility that it is exactly zero, slightly negative, or slightly positive, so for the time being no possibility is excluded.

In another post I mentioned that Dante, in The Divine Comedy, proposes a cosmology that expands Ptolemy’s by adding nine more spheres, those of the Empyrean (the abode of God), and that anticipates Einstein by proposing a finite but unbounded universe, which needs extending our imagination to four dimensions to describe it understandably.

Let’s look at a warning by Carlo Rovelli about the dangers of speculating about infinite space:

I suspect that what we feel as infinity is only the immensity of what we realize that we cannot reach… this is just the sense of our smallness, our frailty, our ignorance.

In the next post I will talk about infinite time.

More information: 

a) Ellis, George F.R. & Brundrit G. B., “Life in the Infinite Universe”, Quarterly Journal of the Royal Astronomical Society 20: 37-41, 1979. 

b) Garriga, J. & Vilenkin A., “Many worlds in one”, Physical Review D 64: 043511-1-5, 2001. 

c) Soler Gil F.J., Alfonseca M., “About the Infinite Repetition of Histories in Space”,  THEORIA, 2014. 

d) Rovelli C., “Some Considerations on Infinity in Physics”, en Infinity: New Research Frontiers, ed. Heller M. & Woodin W.H., Cambridge University Press, 2011.

The same post in Spanish

Thematic Thread about Standard Cosmology: Previous Next

Manuel Alfonseca