So they are directly analogous to time without beginning. 'Infinite' time, if you would.
Time that never finishes.
If each prior moment is represented by a negative integer it is impossible to count all of them before the present moment.
Indeed. It would be silly to even try. But that doesn't mean that the past must be finite; only that if it isn't, it has no beginning - which is fairly obvious if you think about it.
It means infinite prior moments can't have "occurred" before the present moment. Something that never finishes can't have finished before the present moment. It never finishes.
The present moment is not a prior moment so we know all the prior moments must finish before the present moment can occur.
So what? We are, observably, at the present. So all of the past is done. Even if it is infinite.
Absolutely not.
If the number of prior moments to any present moment were infinite then that present moment could never arrive.
The infinite prior moments would never finish for it to happen.
The fact that we experience a present moment means the number of prior moments could not have been without end. They could not have been infinite.
If there are infinite prior moments they can't finish at the present moment. Just like counting the negative integers can't finish at the present moment. It can't finish no matter how much time you have.
You can't finish counting the negative integers, but you CAN start counting them. There is no 'earliest point in time', but there is a most recent point in time. The past has no start, but it ends at the present. Or if you look back from here, it starts at the present, and has no 'end' - ie, it is infinite.
Let the negative integers represent prior moments.
Now start counting.
If they are infinite you will never finish.
If they never finish they can't have finished at any present moment.