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In the run-up to the recent conclave, someone created a diagram consisting of three overlapping circles containing the names of several papabile cardinals. The first circle was labelled “Global Fluency”, the second “Governance”, and the third “Charisma”. In hindsight, it should be no surprise that Cardinal Robert Prevost was at the intersection of all three.
Diagrams of this form, which use circles to depict sets of items, are known as Venn diagrams. For most of us, Venn diagrams are about as far as we get into the subject of sets. However, someone like Pope Leo XIV—who majored in mathematics at university—would have been taught that sets are among the fundamental building blocks of mathematics. Their importance cannot be overstated.
Georg Cantor, a 19th-century mathematician, played a pivotal role in the development of set theory. He was particularly interested in the notion of infinity, and it was this interest that drew the attention of the Catholic Church. Joseph Dauben, a professor of the history of mathematics, gives a fascinating account of Cantor’s interaction with the Church in a paper titled “Georg Cantor and Pope Leo XIII: Mathematics, Theology, and the Infinite”.
In 1879, Leo XIII wrote the encyclical Aeterni Patris, in which he encouraged a renewal of philosophical thought inspired by St Thomas Aquinas. The encyclical was motivated by Leo XIII’s conviction that science could profit from an engagement with Thomism, and in the process the Church could further its own goals and ideals. Cardinal Johann Franzelin and his student Fr Konstantin Gutberlet took up this invitation by engaging with Cantor’s work on infinity.
A central idea in Cantor’s understanding of infinity was his criterion for measuring the size of different sets: if the items of two different sets could be put in one-to-one correspondence with each other, then the two sets were of the same size. For instance, for any whole number, you can double it to get an even number, and for every even number, you can halve it to get a whole number. According to Cantor’s measurement criterion, the set of whole numbers (0,1,2...) and the set of even numbers (0,2,4...) have the same—albeit infinite—size.
But Cantor realised that this criterion implied there could be many different sizes of infinity. He considered how many subsets of the whole numbers there might be. One can easily think of examples: the set of even numbers, the set of odd numbers, the set of prime numbers, or any random increasing sequence. Cantor showed that it was impossible to put the set of whole numbers in one-to-one correspondence with the set of its subsets. He thus concluded that the latter infinite set was larger than the former. He referred to these different sizes of infinity as transfinite numbers.
Many Catholic theologians were hostile to the idea that infinite numbers really existed. Their concern was rooted in the Thomistic understanding of infinity, which held that the only actual infinity was God. To claim there could be actual infinities in the natural order seemed to suggest an identity between creation and the divine—a heresy known as pantheism.
However, if there were many different sizes of infinity, as Cantor claimed, then the existence of actual infinities in creation did not imply identity with God. Though Cantor was raised a Christian, he later said he belonged to no Church or denomination. Nevertheless, his correspondence with Gutberlet and Franzelin increasingly encouraged him to reflect on the theological implications of his theory.
Rather than posing a threat to Christianity, Cantor believed that the reality of transfinite numbers highlighted God’s nature and dominion over the created order. In defending Cantor, Gutberlet argued that since God’s knowledge must be perfect and unchanging, it must include knowledge of transfinite numbers. He concluded that the transfinite numbers must exist in the mind of God.
This did not mean that Gutberlet or Franzelin endorsed Cantor’s view that concrete infinities exist outside the divine intellect. But in a letter to Franzelin in 1886, Cantor made a distinction between the absolute, uncreated infinity that belongs to God alone, and created infinity, which belongs to the transfinite numbers. This distinction satisfied Franzelin that the theory posed no threat to Catholic theology.
Cantor’s conviction that transfinite numbers actually exist in the natural order is still, however, a significant departure from Aquinas’s view. For Aquinas, infinities in nature were only potential—for instance, one can continue dividing a line indefinitely, but not actually divide it infinitely many times.
Regardless of whether one believes in the reality of transfinite numbers, Cantor’s work represents a profound contribution to mathematics. His development of set theory and his deep engagement with the concept of infinity continue to influence contemporary thought.
