to forget: to let someone or something drop out of mind and care, ranging from a simple lapse of memory to deliberate disregard
English readers almost always meet this verb as "forget," but in the New Testament it covers far more than a slip of memory. At one end it is exactly that: the disciples "forgot" to bring bread. At the other end it means turning away on purpose. When Paul says he is "forgetting what lies behind," he is not suffering amnesia; he is refusing to let his past hold him back.
The heavier sense comes from the Old Testament. In the Hebrew Scriptures, to forget God (שׁכח) is not to misplace a memory but to abandon him, the opposite of staying faithful. The Greek Old Testament used this one verb to carry that whole weight, and the New Testament writers inherited it. So "do not forget to show hospitality" does not warn against absent-mindedness; it means "do not let the practice lapse," which is close to the plain word for neglect (ἀμελέω).
Because of that background, the same verb can be gentle or grave depending on the setting. In a story it is an ordinary detail; in a command or a warning it names a failure of loyalty or duty. And when God is the one who does not forget, the point is not merely that he keeps people in mind but that he never drops them from his active care.