Discover more from Beyond Belief
Math and the Mind of God
Georg Cantor and the three levels of existence.
by Joshua M. Moritz
These days we are quite comfortable with the notion of infinity. As scientists discuss the possibility of an infinite multiverse, our entertainment features infinity wars and superheroes who travel to infinity and beyond. Yet, in the past, infinity was not so easy to come by. This is because whenever the greatest thinkers in history pondered the idea of infinity, they soon realized that this concept is riddled with paradox, breaking all the rules of common sense. Logically speaking, infinity didn’t add up.
The Paradox of Infinity
Take, for example, the Achilles Paradox articulated by the ancient Greek philosopher Zeno. If the space or distance between point A and point B can be divided into increasingly smaller segments, then there is no such thing as the smallest distance since there can always be one that is smaller. There are thus an infinite number of smaller segments between point A and point B. But this means that if the Greek hero Achilles wants to run from point A to point B, it would take him an eternity to reach his destination because he has an infinite number of small spaces to traverse. And yet somehow, Achilles can move from point A to point B.
Such paradoxes and contradictions led another Ancient Greek philosopher, Aristotle, to distinguish two different types of infinites. The first is an actual infinite, an infinite that embraces everything in reality, leaving nothing outside.
The second is a potential infinite, which Aristotle describes as a type of infinity that always has something outside of it. In other words, a potential infinite is a situation where you can always add one more number. As Aristotle says, “one thing is always being taken after another, and each thing that is taken is always finite, but always different.”1 However large a finite number you have reached, you can potentially add more, but you can never actually have all of the numbers as a completed set or group. Because you can never actually count to the highest number, Aristotle declared the existence of an actual infinite to be philosophically forbidden.
Like his teacher Plato, Aristotle believed that the actual world was logically constructed and not a realm of contradiction and chaos. Consequently, he believed that there are no actual infinities in physical reality.
From Infinity to God
Still, one actual infinity lurked at the heart of Aristotle’s physics that went unnoticed for almost a thousand years—his idea that the physical world was eternal. The contradiction at the core of Aristotle’s philosophy was first pointed out by John Philoponus, an Alexandrian mathematician, philosopher, scientist, and Christian theologian. Philoponus demonstrated that if the cosmos is uncreated and has no beginning (as Aristotle argued), then an actual infinity of years must have passed. If the world is eternal, then an infinite number of moments must have been traversed. However, if the infinite has been traversed, then it is not truly the infinite. Philoponus used this paradox of infinity to prove that a transcendent God must have created the universe. Philosopher and historian Richard Sorabji explains that Philoponus “found a contradiction at the heart of paganism, a contradiction between their concept of infinity and their denial of a beginning,” and this was a key “turning point in the history of philosophy.”2
The Mathematical Mysteries of Infinity
More than a thousand years after Philoponus, Galileo continued to explore the mysteries of infinity through mathematics. In 1638 Galileo discovered a paradox of infinity as it concerned the set of natural numbers (1, 2, 3, 4, 5…) compared to their squares (1, 4, 9, 16, 25…). A common sense approach would lead one to believe that there are more natural numbers (N) than squares (N2) because not every natural number is a square. However, Galileo set up a one-to-one correspondence between these sets to show that the number of the elements of N is equal to the number of the elements of N2—because there is always a next highest number. In other words, these two sets become equal when infinity is brought into the picture (since they are both infinite). Galileo’s paradox reveals that infinite collections don’t obey the rules of common sense. Even if one subtracts an infinity from an infinity, one still has an infinity left over. Infinity appears to give us something for free without using anything up.
From God to Infinity
It took the genius of Georg Cantor to show how to make sense of the paradoxes of infinity that Galileo had first identified three hundred years before. Cantor founded set theory, discovering that some infinities are bigger than others. Through these discoveries, explains mathematical physicist John Barrow, “Cantor produced a theory that answered all the objections of his predecessors and revealed the unexpected richness hiding in the realm of the infinite.”3 Cantor’s set theory was a revolution in the history of mathematics because it overthrew all the assumptions that previous generations held about infinity.
Today, says mathematician Ian Stewart, “the fruits of Cantor’s labors form the basis of the whole of mathematics.”4 According to Cantor himself, his insights into the nature of mathematics and infinity were revealed to him by God. As he states in a letter from 1883: “I am far from claiming my discoveries are due to personal merit, because I am only an instrument of a Higher Power that will continue to work long after me, just as it revealed itself thousands of years ago to Euclid and Archimedes.”5
To demonstrate how some infinities are larger than others, Cantor introduced the mathematical concept of a “completed set.” For example, Cantor considered the natural numbers (N = 1, 2, 3, 4…) together as a set in themselves, as a completed infinite magnitude. Taken as a whole, Cantor defined the set of natural numbers as the first “transfinite number” (denoted by the lowercase omega, ω), and he deliberately distinguished ω from the infinity symbol, ∞, which had been introduced by John Wallis in 1655 to mean simply “unbounded.”
Next, Cantor defined a countable infinity to be one that can be put into one-to-one correspondence with the list of natural numbers (1, 2, 3, 4, 5, 6, . . .). Thus, for example, the even numbers are countably infinite, and so are all the odd numbers. All countably infinite sets have the same ‘size’ in Cantor’s sense. Cantor understood these to be the smallest infinities that could exist and denoted them by the first letter of the Hebrew alphabet, the symbol Aleph-nought (ℵ0). He revealed that all the fractions formed by dividing one whole number by another are also countably infinite. All the infinities that philosophers and mathematicians discussed in ancient times were countable infinities in Cantor’s sense.
Cantor then set about demonstrating that an unending catalog of mathematical infinities exists. He showed how precise definitions of things like infinite sets lead to the conclusion that ever larger ones can be defined. In his discussion of mathematical reality, Cantor distinguished three levels of existence: (1) the mind of God, (2) the human mind, and (3) the physical universe. “The set of everything” and “the set of all sets” belonged to Absolute Infinity, which is beyond mathematical formulation and which is comprehended only in the Mind of God. Cantor believed that God put the concept of number, both finite and transfinite, into the human mind and that their existence in the Mind of God was the basis for their existence in the human mind. He differentiated between the transfinite numbers, as created in the physical universe and in the human mind, and eternal and uncreated Absolute Infinity, which was reserved for God and his attributes.
While infinites existed mathematically, Cantor did not believe that the universe was infinite in either duration or extent. Cantor firmly believed that God had created the universe in the beginning. The only actual infinity is God. Citing Augustine’s City of God, Cantor affirmed: “All infinity is in some ineffable way made finite to God, for it is comprehended by his knowledge.”6 For Cantor, then, God’s infinity is both the beginning and the end—the alpha and the omega—of all other infinities.
Aristotle, Physics, 3.6.206a.27-29.
Richard Sorabji, Philoponus and the Rejection of Aristotelian Science (Cornell University Press: 1987) 220.
John D. Barrow, The Infinite Book: A Short Guide to the Boundless, Timeless and Endless (Knopf Doubleday , 2007), 67.
Ian Stewart, From Here to Infinity (Oxford University Press, 1996) 67
Georg Cantor, Letter from Cantor to Professor C.A. Valson (Halle January 31, 1886)
Augustine, City of God, 2.238