To prove some basic math facts, you have to use 'infinite' numbers that are so massive they might not even legally exist in standard logic.
April 13, 2026
Original Paper
Free Left Distributive Algebras and a Canonical Extension
arXiv · 2604.08768
The Takeaway
This research shows that even simple algebraic patterns are tied to the most extreme, abstract reaches of the universe. It proves that some basic truths about small objects can only be understood by looking at types of infinity that are too large for our standard mathematical rules to even describe.
From the abstract
Assuming a large cardinal hypothesis, Laver gave a representation of the monogenerated free left distributive algebra (LDA) using elementary embeddings and used this representation to prove many algebraic results. Some of these results were later proved by Dehornoy in ZFC, without the large cardinal hypotheses. However, there is an important algebraic result whose consistency strength is unknown. (See Laver (1995) and Dougherty & Jech (1997).) Recent results [arXiv:2508.02244] extend the connect