November 24, 2025
Set-to over sets
Set theory with types
Math nerds clash: Is everything a set or should we keep things in their lanes
TLDR: De Bruijn’s revived essay argues math should keep objects in their own “types,” not force everything to be a set. The comments split: one side says stop labeling fights and focus on clear definitions; the other loves the minimalist elegance of building all math from one simple framework.
A vintage bombshell just dropped into modern math Twitter: NG de Bruijn’s 1973 paper on “Set Theory with Type Restrictions” resurfaced, poking the bear of “everything is a set.” The essay side-eyes the wild Zermelo–Fraenkel (ZF) world where numbers, points, and pairs are all elaborate sets, and even the nonsense of writing x ∈ x lurks. The vibe? Keep things in their types, stop forcing apples and airplanes into the same basket.
Cue the crowd. One camp, led by epolanski, rolled their eyes at the headlines and went full math teacher: labels don’t matter—definitions do. In other words, stop arguing what to call stuff and just say what it means. On the other side, bananaflag proudly waves the minimalist flag: being able to build all of math from one kind of object with one relation is “relaxing,” like zen for formalists. Fans joked about kindergarten set lessons and “empty set” burns, while meme lords dunked on the ordered pair = {{x},{x,y}} like it’s the world’s nerdiest IKEA hack.
The hottest drama isn’t about who’s right; it’s about vibes: tidy types versus spartan sets. De Bruijn’s old-school critique reads fresh—close the gap between math and meaning—while the comments turn it into a cage match over whether foundations should be clean, clear, or gloriously uniform.
Key Points
- •The article centers on N.G. de Bruijn’s 1973 paper proposing set theory with type restrictions to motivate AUTOMATH.
- •It contrasts Zermelo–Fraenkel (ZF) foundations with typed approaches, highlighting objections to “everything is a set.”
- •Examples of problematic ZF encodings include x ∈ x, Russell’s paradox, ordered pair coding, and naturals-as-sets.
- •Historically, “type theory” often meant higher-order logic before about 1980; today it is linked to dependent type theory in tools like Lean and Agda.
- •Typed sets align with practices in Isabelle/HOL and could be used more broadly in HOL-based proof assistants.