February 23, 2026
Endless hot takes
How Can Infinity Come in Many Sizes?
‘Infinity Comes in Sizes’—commenters demand more wow
TLDR: The piece explains that some infinities are “listable” while the real numbers form a bigger, unlistable infinity. Commenters loved the intro but blasted it for stopping at two sizes, demanding power sets, infinity arithmetic, and answers about “in-betweens,” sparking jokes, nitpicks, and math flamewars.
Infinity isn’t just endless—it’s apparently available in different sizes, and the comments section is having an existential crisis. The article tours Cantor’s wonderland: “small” infinity (you can list things one-by-one like the whole numbers) versus a “bigger” infinity (the real numbers like every point on a line), proved with a digit-flip trick. Cue the chorus: “Only two sizes? That’s it?” Top-voted math fans accuse the piece of stopping at the good part, teasing paradise and then slamming the door. One user points readers toward the big missing chapter: power sets (the set of all subsets), which blow up the size every time you take one.
Defenders clap back: “It’s an intro, not a PhD!” Meanwhile, chaos blooms: a wave of curious newbies asks if there are infinities between the two (hello, the legendary continuum mystery), while pedants spar over decimal quirks like whether 0.999… equals 1 and how to avoid that in the digit trick. Meme squad shows up with Hilbert’s Hotel “no vacancy” jokes, “infinite scroll is uncountable” quips, and one classic: “My inbox is a larger infinity than my free time.” The mood? Enchanted but impatient—the crowd wants the next level: power sets, set-size arithmetic, and the full drama of Cantor’s paradise, not just the lobby. Read the comments; bring popcorn and a number line.
Key Points
- •The article defines sets and cardinality, explaining that counting corresponds to creating a one-to-one match with natural numbers.
- •Even numbers and natural numbers are shown to have the same cardinality via a bijection, illustrating countable infinity.
- •Rational numbers are enumerated by arranging them in a grid and tracing a snaking path, proving they are countably infinite.
- •Real numbers are proven uncountable using a diagonal construction that contradicts any attempt to list them all.
- •The piece concludes that infinite sets can be the same size or different sizes, and there are infinitely many sizes of infinity.