Below is a question and proof that I've done. I was wondering if there is a more formal way of concluding a point must exist that is not in a set composed of a finite number planes. Currently I am using an informal argument based on the idea that such a set would have no "volume". Additionally, how would I / could I extend my induction argument to prove this true not only for graphs with a finite number of members but also for a countably infinite number of members?
Can all graphs with a finite number of members be represented in three dimensions using straight lines to represent edges and points to represent vertices without any intersections between edges or edges and points?
Let P be a collection of points with a finite number of members
Let a point q be a member of S if there exists 3 points in P that form a plain with q
The “volume” of S is zero therefor there exists a point not in S
Therefor, for any P there will always exist an additional point q that can be added to P without creating intersecting edges
Therefor, by induction, all graphs with a finite number of members can be represented in 3 dimensions using straight edges
Also, please point out any problems with the proof and how to fix them.
Thanks