Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 144 / 1

Expand the language by constants for every element of $M$ and a new tuple $c$. Let $\mathrm{ElDiag}(M)$ be the elementary diagram of a structure associated with $M$. Every finite subset ofis realized in the expansion of $M$, because the finitely many formulas from $p$ are simultaneously satisfiable in $M$. The compactness theorem gives a model $N^{\prime }$ of the whole set. The interpretations of the named constants give an elementary embedding $M\to N^{\prime }$, and the interpretation of $c$ realizes $p$. Identifying $M$ with its image produces an elementary extension $M⪯N$ realizing $p$. This is the finitely satisfiable type is realized in an elementary extension argument.

Write $\kappa =|\mathcal{U}|$ and define on tuples realizing $\mathrm{tp}(\bar{b}/\varnothing )$ the definable equivalence relationThe $E$-classes correspond exactly to the members of $O(\mathcal{D})$, because automorphisms of the monster model carry $\bar{b}$ through all realizations of its type.

There are at most $\kappa$ such tuples, so $|O(\mathcal{D})|\le \kappa$. If the orbit is infinite, recursively choose ${\bar{b}}_{\alpha }$ for $\alpha <\kappa$ in pairwise different $E$-classes. At stage $\alpha$, the typeis finitely satisfiable because there are infinitely many classes, and its parameter set has size less than $\kappa$. Saturation realizes it in $\mathcal{U}$. Thus there are at least $\kappa$ classes, proving the orbit of a definable set in a monster model identity