A theory is <kappa-satisfiable theory>[$\kappa$-satisfiable] when every subtheory of size below $\kappa$ has a model. An uncountable cardinal $\kappa$ is <weakly compact cardinal>[weakly compact] when every $\kappa$-satisfiable theory in an infinitary language $L_{\kappa,\kappa}$ with at most $\kappa$ nonlogical symbols is satisfiable.

Equivalently, $\kappa\to(\kappa)^2_2$: every coloring $c:[\kappa]^2\to2$ has a homogeneous subset of cardinality $\kappa$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 116 / 1 / a / ii / Solution

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An uncountable cardinal $\kappa$ is weakly compact when every $\kappa$-satisfiable theory in an $L_{\kappa,\kappa}$ language having at most $\kappa$ nonlogical symbols is satisfiable.


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 Incoming links: Weakly compact cardinal