*Infiniti is back. Or rather, it never disappeared (never, never…). While mathematicians have a good sense of infinity as a concept, cosmologists and physicists find it more difficult to understand infinity in nature, writes Peter Cameron. *

Each of us has to face a moment, often rather early in his life, when we realize that a loved one, who was once an essential element of our life, was not infinite, but left us, and that one day we will also have to Leave this place.

This experience, perhaps as much as the experience of looking at the stars and wondering how far they go, shapes our views of infinity. And we urgently want answers to our questions. This has been the case ever since, two and a half thousand years ago, when Malunkyputa put his doubts to Buddha and demanded answers: among them he wanted to know whether the world was finite or infinite, and whether it was eternal or not.

We’ve recently heard John Donne’s words that promise that eternity consists of

*“No noise, no silence, but one music.*

*No endings or beginnings, but one eternity.”*

Hard to imagine, and one music would surely become equally unbearable!

There are many approaches to infinity through the twin pillars of science and religion, but I will limit my attention here to the views of mathematicians and physicists.

Aristotle was one of the most influential Greek philosophers. He believed that we can think of “possible infinity” (we can count things without knowing how many things are coming) but “complete infinity” is a taboo. For mathematicians, infinity was banned for two millennia after Aristotle was banned. Galileo tried to address the problem, stating that an infinite set could be matched to a part, but eventually relented. It was left to Cantor in the 19th century to show us how to think about infinity, which is what most mathematicians now accept. There is an infinite number of counting digits; Any number you type is a negligible step along the way to infinity. So Cantor’s idea was to imagine that we have a package that contains all of these numbers; Label it “Natural Numbers”, and treat the package as a single entity. If you want to study the odd numbers, you can break the package and take it out to look at them. You can now take any combination of these packages and combine them to form one more entity. Thus, set theory was born. Cantor researched ways to measure these sets, and today set theory is the most common basis for mathematics, although other grounds have been suggested.

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If you toss a coin 100 times, it is not impossible (very unlikely) that it will collapse each time. But, if you can imagine throwing a coin infinitely often, the chance of not getting heads and tails evenly is often zero.

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One of Cantor’s discoveries is that there is no largest infinite set: given any set, you can always find a larger set. The smallest infinite set is the set of natural numbers. What comes next is a mystery that cannot be solved at present. They may be real (decimal) numbers, or they may not be. Our current foundations are not strong enough, and building larger telescopes will not help solve this question. Perhaps in the future we will adopt new foundations of mathematics that will solve the question. But for now, since mathematics is a mental construct, we can decide whether or not the universe we’re playing with satisfies the “continuity hypothesis.”

These questions keep the group’s theorists awake at night; But most mathematicians work near the bottom of this dizzying hierarchy, with little infinities. For example, Euclid proved that prime numbers “go on forever”. (Aristotle might say, “Whatever prime number you find, I can find a larger number”; Cantor would simply say “the set of primes is infinite.” Mathematicians (including this year’s Fields Medalist James Maynard of Oxford) seem to be on The twin prime conjecture is nearing completion.Double primes are pairs of primes, such as 3 and 5, or 71 and 73, that differ by only 2. The conjecture, which has not yet been proven, asserts that there are an infinite number of them. But this is the infinity of the natural numbers, the smallest infinity.

Suggested reading

Physics alone cannot answer the big questions

by SabineHossenfelder

Whereas Kronecker (a fierce opponent of Cantor’s ideas) believed in the nineteenth century that “God created the natural numbers; the rest is the work of man”, we can now construct the natural numbers using the tools of set theory, starting from nothing (more precisely, the empty set).

Mathematicians know, however, that there is a huge gap between the finite and the infinite. If you toss a coin 100 times, it is not impossible (very unlikely) that it will collapse each time. But, if you fancy tossing a coin infinitely often, the chance of not getting a shot and tails evenly is often zero. Of course, you can’t really do this experiment; But mathematics is a conceptual science, and we are happy to accept this statement on the basis of rigorous proof.

Infinity has not been satisfactorily resolved in physics and cosmology. The two great theories of physics of the twentieth century, general relativity (the very big theory) and quantum mechanics (the very small theory) have resisted attempts to unify them. The one thing most physicists can agree on is that the universe originated a finite time (about 13.7 billion years ago) – big, but not infinite.

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They deny that infinitesimally small can exist in the universe, but they describe a minimum possible scale, which is essentially the so-called Planck scale.

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The James Webb Space Telescope is just beginning to show unprecedented detail in the universe. In addition to nearby objects, it sees the farthest objects ever observed. Since light travels at a finite speed, these are also the oldest objects observed, having formed near the beginning of the universe. The finite speed of light also puts limits on what we can see; If an object is so far away that its light cannot reach us if it travels for the life of the universe, then we are not aware of its existence. So Malunkyaputta’s question about whether the universe is finite or infinite is moot. But is it eternal or not? This is a real question, and it has not been resolved yet.

Attempts have been made to reconcile relativity with quantum theory. Today’s most promising adopt a very radical attitude toward infinity. They deny that infinitesimally small can exist in the universe, but they describe a minimum possible scale, which is essentially the so-called Planck scale.

Proposed offer

Infinite puzzle

With David Malone, Laura Mercini Hutton, Peter Cameron, Julian Barbour

Such a solution would put an end to the Zeno paradox. Zeno denied the possibility of movement, because to get from A to B, you first have to go to point C halfway to B, and before that to point D halfway from A to C, and so on to infinity. If space is not infinitely divisible, then this infinite regression cannot occur. (Democritus and the early Greek atomists already understood this solution.)

Of course, this leaves us with a conceptual problem similar to the one raised by the possibility that the university is finite. In this case, the obvious question is “If the universe has an advantage, what is beyond it?” In the case of Planck length, the question would be “Given any length, however small, why can’t I just take half of it?”

Perhaps because we have been influenced by Zeno’s paradox, we tend to think of the points on the line to be, like real numbers, infinitely divisible: between any two we can find another. But the current thinking is that the universe was not built this way.

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Time, however, is still a problem

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More importantly for physics, the atomic hypothesis also gets rid of another annoying occurrence of infinity in physics. Black holes in general relativity are points in space-time where the density of matter becomes infinite and the laws of physics break down. This has been a thorn in the body of cosmologists since their existence was first predicted, because by definition we can’t understand what’s going on there. If space is separated, we cannot put infinitely many things near each other, and the paradox is avoided. We can still get very high density; The recently observed and photographed black hole at the center of our galaxy (according to this theory) is just a point of such high intensity that light cannot escape, but it does not challenge our ability to understand it.

However, time is still an issue; Current theories cannot decide the ultimate fate of the universe. Does it end in heat death, a cold, dark universe where nothing happens? Will the mysterious ‘dark energy’ become so powerful that it is tearing the universe to shreds? Or is the expansion from the Big Bang going in the opposite direction, so that the universe ends in a major crisis?

Suggested reading

The Big Bang didn’t happen

Written by Eric J. Lerner

The physicists of the nineteenth century who developed the science of thermodynamics noted that over time, a system as complex as the universe becomes more and more turbulent. (We say that its entropy is increasing.) It has recently been suggested that this is inverted; It’s the increasing turbulence in the universe that somehow causes the passage of time. This is part of a movement in which the traditional units of space, time, matter and energy are being replaced by information as the primary currency of the universe. But this is the early days of such theories.

None of this matters to us individually. The Sun will expand and swallow the Earth long before the universe reaches its end. But we have an insatiable curiosity to know the answer to Malunkyaputta’s question. As mathematician (and optimist) David Hilbert said, “Wir mussen wissen, Wir werden wissen” (We should know; we’ll know.)

*references** *

*Apostolos Doxiadis, Logicomix, Bloomsbury, 2009.*

*Carlo Rovelli, Reality Is Not What It Seems, Riverside Books, 2017.*