Anuj Dawar poses two questions which give finitary analogies to the Löwenheim–Skolem theorems. Grohe , has shown that the first of these, which corresponds to the downward Löwenheim–Skolem theorem, has a negative answer. In this paper we combine Grohe's technique with that of Robinson's famous paper ) to show that the second question, which corresponds to the upward Löwenheim–Skolem theorem, also has a negative answer.