No, O is defined as the point at infinity.
So, we are aiming for the number of times a point can be added to itself so the slope is infinite? i.e. if you looked at a simplified graph, the slope of the line is vertical, or nearly vertical. Or the order is some way you can detect the number of times a point addition can be done to itself before you hit O (infinity). Is the order selected after a number of point additions, or is there an algorithm that gives you an order after repeated point doubling? (Is that the same as adding a point to itself?)
No no no,
you are mixing concepts of the real word with the finite field.
In the real world there is NO infinity (in elliptic curve calculations) You can forever keep doing the additions, and you will NEVER reach infinity.
In the finite field you will reach infinity, when you have to divide by 0 (in the equations).
The order is just the number of different points in the finite field. When you do additions in the finite field, you will go through them one by one, until you have gone through all of them. the last point will be infinity, because you have to divide by zero.
In finite fields, no matter what is your generator point, the point of infinity is the last point, after that the calculations start from the beginning.
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Edit:
In numerical terms, the infinity always happens when the 2 points you are trying to add together are G and -G, two points with the same x coordinate. When calculating the slope s=(y2-y1)/(x2-x1), the x values are the same, and you cant divide by 0
In real world this cannot happen, as there is no point X in the curve that would get you to -G when adding G to X.