If two planes α, β have a point A in common, then they have at least a second point B in common.
I perfectly understand the axiom, but i don't see why it's necessary, and it's kind of counterintuitive for me.
When I think of two planes, I can think of these ways they could intersect:
But I could also think of two planes just "barely" touching in one point, kinda' like this
I know that this is an axiomatic system, so this is just a "a rule of the game", and that it doesn't want to describe what we understand as physical flat surfaces, or what I think of a plane. However, I feel a little werwerfasdre about why this axiomatic system does not contemplate a case in which two planes intersect or touch in just one point. Also, I wonder why this axiom correctly describes our idealization of planes in order to study them, and what was the reason for Hilbert (hence, probably Euclid) to contemplate just two cases in which planes intersect. Could Euclidean Geometry be studied without this axiom?
Thanks in advance