The <compactness theorem> for <first-order logic> says that a set $\Sigma$ of <first-order sentences> has a model if and only if every finite subset of $\Sigma$ has a model. It applies to arbitrary languages, including expansions by arbitrarily many new constants. Equivalently, if $\Sigma$ entails a sentence, some finite subset already entails it. We will use both formulations with the <diagram of a structure>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 23 / 1 / a / Solution

 Solution (source code)