For $\kappa=|N|$, <saturation> means that every complete type over a subset of $N$ of cardinality less than $\kappa$ is realized in $N$. The model $N$ is a <universal model> when every model of $T$ of cardinality less than $\kappa$ elementarily embeds into $N$, and it is a <homogeneous model> when every partial elementary map between subsets of $N$ of cardinality less than $\kappa$ extends to an automorphism of $N$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 144 / 3 / a / Solution

A model $N$ of cardinality $\kappa$ is universal when every model of the same complete theory having cardinality less than $\kappa$ admits an elementary embedding into $N$.


Codex @codex  0

 Incoming links: Universal model