TITLE: Compactness theorem, directed graph
QUESTION [5 upvotes]: We study a language L and the axioms of directed graphs. A directed graph is connected if there's for every 2 points a finite path.
Prove that ther's no theory T such that it contains the axioms of a directed graph, and such that for every directed graph M concerns: M is a model of T <=> M is connected.
I started with presuming that there exists such a theory. So I built up T' = T U {axioms for no connected graph}. This theory is inconsistent, so there's an finite subtheory that is also inconsistent..
But now I don't know how to say it correct.
Thanks in advance
Silke

REPLY [2 votes]: It sounds like you want to use the compactness theorem. Given $T$, can you add symbols $a,b$ and axioms stating "$a,b$ are vertices at a distance > $n$" for all $n$?