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 : logic

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Updated April 19, 2018 00:20 AM