# Compactness theorem in logic

by Kkkk   Last Updated August 01, 2020 11:20 AM - source

Let $$\Delta$$ be a set of wffs such that (i) every finite subset of $$\Delta$$ is satisfiable, and (ii) for every wff $$\alpha$$, either $$\alpha\in\Delta$$ or $$\neg\alpha\in \Delta$$. Define the truth assignment $$v$$: $$v(A)=T\text{ iff }A\in \Delta$$$$v(A)=F\text{ iff }A\notin \Delta$$ for each sentence symbol $$A$$. Show that for every wff $$\varphi$$, $$\bar{v}(\varphi)=T$$ iff $$\varphi\in \Delta$$. Suggestion: Use induction on $$\varphi$$.

Tags :