The Kalam Argument and Counting Backward to Infinity

One of the philosophical arguments used to show the impossibility of an infinite past, per the Kalam Cosmological Argument, is that if an immortal being counted an infinite number of events we could never find her counting if we counted events backward in time. For no matter how many events we counted, the immortal being would already have finished her count to infinity. I've diagrammed it here. So let's suppose an immortal being has been counting events from the beginningless past. Then it's true that no matter how far backward in time we counted events from the present, she would already have finished counting an infinite number of events. However, there is nothing in the Kalam argument that forces us to think the immortal being stopped counting events. If she continued counting into the present then yes, we could find her still counting events. She could continue counting events into the infinite future too. So unless there is a reason to think otherwise this particular argument fails to show anything about whether the past is finite. It might be. It might not be. But this philosophical argument is irrelevant to establishing the case needed.