Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 128 / 1 / b / ii / Solution

The language of set theory has only countably many 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.