Some infinities are bigger than other infinities

Some infinities are bigger than other infinities is a philosophical and mathematical metaphor that Hazel Grace Lancaster uses in her eulogy at Augustus Waters's prefuneral to articulate the idea that the finite time they shared, though brief, constitutes a meaningful infinity within the larger, unbounded set of possible time. The phrase originates from Peter Van Houten's explanation of Georg Cantor's set theory during a drunken monologue in Amsterdam, where he uses the tortoise paradox to illustrate that some infinite sets are larger than others. Hazel later repurposes the concept to express gratitude for her relationship with Augustus, transforming a mathematical abstraction into a deeply personal statement about love, loss, and the value of limited time.

Origin in Van Houten's Monologue

The phrase first appears in the novel during Hazel and Augustus's disastrous visit to Peter Van Houten's Amsterdam home. When Hazel presses him for answers about what happens to the characters after the end of An Imperial Affliction, Van Houten deflects by launching into a lecture about Zeno's tortoise paradox. He explains that in a race between a runner and a tortoise with a head start, the runner can never mathematically catch the tortoise because each interval of distance creates a new, smaller interval. Van Houten then states that "no one really solved it until Cantor showed us that some infinities are bigger than other infinities." He presents this as a non sequitur, claiming it answers Hazel's question, though she finds it unhelpful. Later, in the mirrored elevator of the Hotel Filosoof, Hazel mockingly repeats the line in a drawl, imitating Van Houten's drunken delivery, calling him "an assclown" for his pretentious evasion.

Hazel's Transformation in the Eulogy

Eight days before Augustus's death, at his self-arranged prefuneral in the Literal Heart of Jesus, Hazel delivers a eulogy that redefines the phrase. Standing at a wooden lectern, she begins by stating that she cannot tell their love story because it will die with them, as all real love stories should. Instead, she turns to mathematics. She explains that there are infinite numbers between 0 and 1—0.1, 0.12, 0.112, and an infinite collection of others—but there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million. "Some infinities are bigger than other infinities," she says, crediting "a writer we used to like" for teaching her this. She then directly addresses Augustus: "There are days, many of them, when I resent the size of my unbounded set. I want more numbers than I'm likely to get, and God, I want more numbers for Augustus Waters than he got. But, Gus, my love, I cannot tell you how thankful I am for our little infinity. I wouldn't trade it for the world. You gave me a forever within the numbered days, and I'm grateful." In this speech, Hazel transforms Van Houten's abstract mathematical concept into a metaphor for the value of a finite but meaningful relationship, asserting that the quality of shared time matters more than its quantity.

Thematic Resonance and Recurrence

The idea of relative infinities echoes earlier moments in the novel. When Augustus dives onto a virtual grenade in Counterinsurgence 2: The Price of Dawn to save fictional schoolchildren, he tells Hazel, "All salvation is temporary. I bought them a minute. Maybe that's the minute that buys them an hour, which is the hour that buys them a year. No one's gonna buy them forever, Hazel Grace, but my life bought them a minute. And that's not nothing." This logic—that a small unit of time can have disproportionate value—prefigures Hazel's later mathematical metaphor. Similarly, when Hazel argues with Augustus about the ghettoization of scrambled eggs, she rejects his claim that fragility and rarity make something beautiful, saying, "You're arguing that the fragile, rare thing is beautiful simply because it is fragile and rare. But that's a lie, and you know it." Yet in her eulogy, she embraces a version of this very idea: the brief time with Augustus is precious precisely because it is finite and rare. The swing set advertisement they write together also touches on this theme: "No matter how hard you kick, no matter how high you get, you can't go all the way around." The phrase "some infinities are bigger than other infinities" thus becomes the novel's central resolution of the tension between the desire for more time and the acceptance of limited time.

Significance as a Motif

The phrase functions as a counterpoint to the novel's recurring motif of oblivion. Early in the story, Hazel tells Augustus at Support Group that all human achievement will eventually be forgotten, and that the inevitability of oblivion should be ignored. Augustus later confesses his fear of oblivion, worrying that he will not leave a meaningful mark. Hazel's eulogy offers an alternative: rather than seeking a grand, immortal legacy, one can find meaning in a "little infinity"—a finite but deeply felt connection. The phrase also echoes Augustus's earlier line to Hazel, "You gave me a forever within the numbered days," which Hazel incorporates into her eulogy. By repurposing Van Houten's mathematical lecture into a declaration of gratitude, Hazel asserts that love, not fame or legacy, is the true measure of a life's significance. The phrase becomes a testament to the idea that even a short life can contain an infinite amount of love.