Can you ever really forgive, if you can't forget?

Season 4: Episode 7: Time and Punishment
Carrie and Aidan are interrupted in bed by the voice of Big on Carrie's answering machine. Though Aidan says nothing, Carrie fears that he'll never be able to forgive her for her affair with Big. She wonders if you can ever really forgive if you can't forget. Carrie works hard to show Aidan that she's a great girlfriend, but he seems distant and angry.

"Think how much easier it would all be if there were some swift surgical procedure to whisk away all the ugly memories and mistakes and leave only the fun trips and special holidays. But until that day arrives, what to do? Rely on the same old needle point philosophy of forgive and forget? And even if a couple can manage the forgiveness, has any ever really conquer the forgetness? Can you ever really forgive, if you can't forget?" - Carrie

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Is it possible? Hmm. My answer: It would be possible to genuinely forgive but it would be impossible to totally forget. What about you? Tell me.