A large portion of the theory of large cardinals—beyond measurable cardinals—focuses on studying the structure of elementary embeddings of the universe of sets into inner models. This talk will address a fundamental question in this context: Does the inner model uniquely determine the elementary embedding?

In this direction, we will present one of the main results of [1], namely:

Theorem 3.5. Any pair of elementary embeddings of the universe into the same inner model agree on the ordinals.

[1] GOLDBERG G. THE UNIQUENESS OF ELEMENTARY EMBEDDINGS. The Journal of Symbolic Logic. 2024;89(4):1430-1454. doi:10.1017/jsl.2024.60