> The set of points is defined to be not collinear, thus we cannot draw a line passing through more than two of them This is wrong and you are misunderstanding

> The set of points is defined to be not collinear, thus we cannot draw a line passing through more than two of them This is wrong and you are misunderstanding what collinearity means. You could have a set of points where all but one are on the same line, and the set of a whole will be not collinear, while we can obviously draw a line that passes through more than two of them. A different way of stating the theorem is that any finite set of points has either a line passing through all points (i.e. the set is collinear) or there exists a line that passes through exactly two points. This dichotomy (why two and not three? Why can't we construct a set where any line passes through at least three points?) is not immediately obvious.