I'm studying for a logic exam and I've become stuck on this problem. My instinct is that it isn't possible but I don't really know how to approach proving why,
Is it possible to express the following in $L$ using a finite set of sentences: $R(x, y)$ is a linear order with left endpoint $b$ and right endpoint $c$ and there are a finite number of distinct objects between $b$ and $c$ in this order?
You can express the property of being a linear order with certain endpoints, but, as the comment of Eric Wofsey suggests, it is not possible to write a finite set of formulas that says something like "there are finitely many distinct elements such that...". This follows from the Compactness Theorem: if you have a set of formulas that have models of arbitrarily large finite cardinality then you have an infinite model of that set of formulas.