Our theme for the month of September is Alphabet Soup. Each writer was assigned a letter and will title their post “___ is for ___.”
German philosopher David Hilbert theorized a Grand Hotel, a lodging with infinitely many rooms hosting infinitely many guests, each guest filling one room. If another guest shows up, they must be out of luck, because each room is occupied. Well, no, Hilbert reasoned; the rooms are infinitely many; the hotel cannot run out of rooms. To accommodate the extra person, each current guest of the hotel must move up one room: from room 1 to 2, room 9 to 10, room 139510 to 139511. The newcomer gets a cushy bed in room 1.
The problem goes on, explaining how the hotel could also accommodate a coach filled with infinitely many new guests, or infinitely many coaches each filled with infinitely many new guests, and after a while the non-mathematical mind starts to agree with Douglas Adams in The Ultimate Hitchhiker’s Guide to the Galaxy: “Infinity itself looks flat and uninteresting.” With so many infinities—infinitely many possibilities of infinity + 1 and n times infinity and infinityn—the infinitely fascinating concept of infinity might become dry, still incomprehensible but inspiring more eye-rolls than awe.
Some 2,200 years before Hilbert’s Hotel, Jain mathematicians made a simpler, and perhaps more enjoyable, distinction of infinities: asaṃkhyāta and ananta. Asaṃkhyāta (in some versions asaṃkhyeya) is countless, innumerable, incalculable—a rigidly bounded infinity. In the Torah, God tells Abraham, “I will make your offspring as numerous as the stars of heaven and as the sand that is on the seashore”; his descendants will be too many to count. Ananta is endless, unlimited—a loosely bounded infinity. In ancient Egypt, the gods Ḥeḥ and Hauhet personified the infinite. “Ḥeḥ” meant “flood,” referring to the infinite chaos that preceded the creation of the finite world.
Humans uncritically romanticize the infinite, yet the concept of larger and smaller infinities seems to permeate even without a theoretical foundation. “Some infinities are bigger than other infinities,” Hazel says as she memorializes Gus and their “little infinity” in John Green’s The Fault in Our Stars. Gustave Flaubert alludes to the same in Madame Bovary, the titular Emma discovering how “an infinity of passion can be contained in one minute, like a crowd in a small space.” Even Buzz Lightyear plans to go “to infinity and beyond!”
One day, our sun will run out of hydrogen in its core, causing the star to expand so far that it swallows the earth, extinguishing all. Eventually, not just our local star, but our entire galaxy will decay until it reaches the background temperature of the universe at 2.73 Kelvin (-270.4°C or -454.76°F). On the cosmic scale, our little infinities become meaningless. All that we might consider infinite on this planet—renewable energy, the reaches of the mind, the potential of humankind—will end.
Some infinities are certain: the infinitely many numbers between 0 and 1, the infinitely many digits in an irrational number, the infinite energy that exists in the universe. Others we will never know. The size of the observable universe is finite, but is the universe itself infinitely large? Time appears to be finite—we can trace it back 13.82 billion years, with our earliest sights on the cosmic background radiation that became visible 380,000 years after the Big Bang (alternative models, for the present, excluded)—yet we do not know if time preceded the universe, or why time moves forward, or if time will end. In Christian eschatology, heaven and hell are eternal (temporally infinite); the same is true of hell and Paradise in Islam. Yet to reach any of these places, one must first die, rendering our living knowledge of them finitely limited.
Pondering infinity serves little practical purpose in our finite lives, yet the concept thrills and inspires. Young children, after first learning about infinity, might ask a genie for “infinity wishes”; aspiring scientists may be invigorated by the seemingly infinite blades of grass or members of an ant species or layers of sediment found on earth; those of religious conviction may be comforted (or terrified) by the apparently infinite nature of the soul. One might poetically argue that humans are infinitely fascinated by infinity, even if, as Adams writes, the fact of infinity is far less interesting than our explorations.