Elliot Glazer
@ElliotGlazer
Followers
3K
Following
17K
Media
66
Statuses
1K
Set theorist and AI math benchmarker. Researches the axiom of choice. Led the development of FrontierMath.
Berkeley
Joined November 2024
Before the labs got IMO Gold, I watched models 10x the state-of-the-art on FrontierMath, the benchmark I planned to last 5+ years. I have no more doubt: AI will radically reshape mathematical research in the coming years. I'm done being a referee. I've joined Principia Labs, a
36
50
841
Just downloaded two 30-year old math papers from the public access terminal in the bowels of UC Berkeley’s library, as God intended
7
1
92
tfw i set out to post one example of a choiceless pathology (an infinite boolean algebra with no infinite antichain) and end up finding a full characterization where none had been expected
1
0
11
See https://t.co/zyNf8zI0zr for an elaboration of the second model.
0
0
4
Open Question 1: No, take M=\bigcup_{\alpha<\omega^2} L(P(\{a_{\beta}: \beta<\alpha\})), for atoms a_{\beta}. Open Questions 2 and 3: No, take the plenitudinous extension of N=\bigcup_n L(\bigcup_{i<n} \{a_{n, \beta}: \beta<\omega_{n+1}\}^{\omega_n}).
Bokai Yao: Plenitudinous Urelements and the Definability of Cardinality https://t.co/b2AHPgCfVd
https://t.co/uM94UJZwXM
2
0
10
I need to stop talking about the axiom of choice at SF parties…
I refuse to believe you can just select an element from any arbitrary set. That’s hubris. That’s Biblical levels of greed.
4
0
60
MathArena goes visual: We evaluated models such as GPT-5 on Math Kangaroo 2025, a recent contest for ages 6-19 where most tasks require visual reasoning. Models struggle the most with tasks for younger kids. For example, they get this task for 1st graders only 3% of the time 🧵
2
19
71
Conclusion of the lattice talk at my math+tacos party: “X is Beyond Meat iff X meet meat = meat.”
1
0
10
Back when I only knew how to get positive results in ZF, I tried proving you *could* deduce X has a partition into 2 (and many more) sets. Guess I don’t have to feel guilty for being unsuccessful!
0
0
6
If there is a non-injective surjection from P(X) to itself, clearly X is infinite. Proven here: without the axiom of choice, one cannot deduce that X has a partition into *two* infinite sets.
Hu, Mao, Shen: Amorphous sets and dual Dedekind finiteness https://t.co/jwmAqUjtmS
https://t.co/XdtffNT1rr
1
1
15
Open question in a recent 100+ page paper: I knew a theorem settling it for all ∞>p>1. GPT-5 Pro independently surfaced another paper proving it for ∞>p>2, a reference I’d missed. Partial, yet genuinely impressive.
4
18
251
We manually evaluated three compute-intensive model settings on our extremely hard math benchmark. FrontierMath Tier 4: Battle Royale! GPT-5 Pro set a new record (13%), edging out Gemini 2.5 Deep Think by a single problem (not statistically significant). Grok 4 Heavy lags. 🧵
24
67
566
We automatically generated the unit test that would’ve caught @AnthropicAI’s top-K compiler bug without relying on their bug reproducer code. Most testing pipelines never hit rare bugs until they fail in production. Ours do.
2
8
49
Without the axiom of choice, you can't even prove such a space has a countably infinite subset!
Them hoes was tryna figure out if erry compact Hausdorff space wit at least two points and no isolated points got a cardinality of at least 2^ℵ₀. 😸😸😸
0
0
12
With the development of fine structure theory, we understand the minimal model of set theory to a similar “resolution” as the standard model of arithmetic. Each theory is undecidable, but in both cases, we’re as close as possible to a well-specified universe.
0
0
16
Hot take: if ZFC is found inconsistent, Peano Arithmetic will almost certainly fall within a year. ZFC is equiconsistent with ZFC + V = L, and replacing Inf with its negation gets PA. The theories are structurally extremely similar.
@Kropotkin57 Lean proves the consistency of ZFC so it would instantly prove all propositions if ZFC is found inconsistent. And the hope that other mathematicians or programmers could retreat to weaker foundations depends heavily on the axioms used in this hypothetical contradiction.
9
3
94
The great Michael Atiyah gave an interview on the future of mathematics (back in 1997!) Each generation develops on the mathematics of the previous, with more and more results of increasing complexity "And it's a miracle that it keeps happening!"
1
1
16
There's an old rumor my PhD advisor Hugh Woodin long ago discovered an inconsistency in ZFC and his frequent genius discoveries are obfuscations of that proof...
@samth @doomslide I’ll just say it would be a huge deal if Lean’s type theory is inconsistent (I believe it’s below ZFC + infinitely many inaccessibles, so a “mild” foundation). RL finding such an inconsistency would be by far the greatest math discovery by AI!
15
8
227