About Affine planes

Question

I am studying about affine planes An affine plane can be defined as It is an ordered pair ($\mathcal{P}$ ,$\mathcal{L}$), P is non-empty set of points and and L is non-empty collection of the subsets of $\mathcal{P}$ called lines satisfying the following properties

1. Given any two points, there is a unique line joining any two points.

2. Given a point P and a line L not containing P, there is a unique line that contains P and does not intersect L.

3. There are four points, no three of which are collinear. (This rule is just to eliminate the silly case where all of your points are on the same line.) Now what is the problem and what I am thinking about affine plane is the example of Rational affine plane where $$\mathcal{P}=\{(x,y)/x,y\in Q\}$$ and $$\mathcal{L}=\{(x,y)/ ax+by=c\}$$ But in this case the each line in rational affine plane seems to dots. means line is discontineous. Similalry in case of the finite affine planes I am thinking. Can anyone help to remove my this confusion. Thanks in advance

Answer 1

You are correct that in either a finite affine plane (like the affine plane with just four points) or the rational affine plane, there's a "natural" way to put the plane within the usual Euclidean plane, and when you do so, the plane is "disconnected".

What this shows you is that an affine plane, as defined by these simple axioms, doesn't have all the properties that you might expect of the Euclidean plane. One possible conclusion is that someone chose the wrong axioms; a different one is that these axioms are rich enough to define a class in which many interesting theorems are true, and that there's a strong enough analogy with the familiar plane to make it worth studying. (As you may guess, I favor the latter view.)

You can regard this as being similar to complex numbers: they satisfy many of the same axioms as the real numbers, but if you try to put an "order" on them (a way to say that one of them is "less than" another) that has all the properties of the usual less-than order on the reals, you cannot do it. You could say either "OK, the complex numbers are stupid and not worth studying" or "OK, well, I guess I won't be able to use all the things I know about order, but at least square roots will make more sense!" :)