Why can't reachability be expressed in first order logic?

Solution 1:

The issue here is with the terminology. You can express reachability in first-order logic; for example you can express it in the language of set theory by quantifying over paths in the graph, and set theory is a first-order theory.

You cannot express reachability in the particular language where the only relations available are the incidence relation on the graph and equality, and where quantification is only permitted over elements of the graph. So the issue actually has very little to do with "first order logic", and everything to do with the particular language that is being allowed. Tradition says that this situation is phrased as "reachability cannot be expressed in first order logic", even though that phrase could be interpreted in other ways.

Ignoring terminological issues, it can be genuinely useful to know that reachability cannot be expressed without quantifying over sets or paths in some way. For example, that fact means that if your graph is stored in an SQL database in a natural way then it will be more difficult to code an SQL query that tells whether an arbitrary pair of nodes has a path than it would be if that relation could be phrased in the limited language mentioned above.

By the way,here is how you show there is no formula $\phi(a,b)$ expressing reachability. Take the language of graph theory mentioned above and add two new constants $C$ and $D$. Take the infinite set of axioms "there is no path of length $1$ from $C$ to $D$", "there is no path of length two from $C$ to $D$", ... Then any finite set of these axioms is consistent with $\phi(a,b)$, as can be seen by looking at a graph with an infinitely long path and taking $C$ and $D$ to be far enough apart. Thus by the compactness theorem the set of all those axioms is consistent with $\phi(C,D)$. But if all those axioms are true in a graph, there is no path from $C$ to $D$ in that graph, so $\phi$ did not express that property correctly.