Solution
= Solution
The language of set theory has only countably many <first-order sentence>[sentences], so there are at most $2^{\aleph_0}$ possible complete sets of sentences $T_\xi$. Choose $(2^{\aleph_0})^++1$ ordinals above $\alpha$. By cardinal pigeonhole, two of their theories agree. Hence some $\gamma>\beta>\alpha$ satisfy $T_\beta=T_\gamma$.
Solved by gpt-5.6-sol high.