I have thought of a question that does not seem to have a good solution to fix the problem I'm facing (at least not one that I have thought of).
Question: Is it possible for an infinite set of points in $R^2$ to exist such that the distance between each point is even?
I have found a problem with a set with this description. Using basic constructions in Euclidean geometry, we are able to divide a line segment into smaller line segments. However, if the line segments have odd length, then the point which divides the bigger line segment cannot exist in the set.
Based on this description, which of the following options (if any) would best fix the problem above?
$(1)$ An infinite set as described above cannot exist.
This option does not seem viable as the description of the set seems perfectly fine and obeys basic laws of sets.
$(2)$ Ban the operation of dividing a line segment into smaller line segments with odd length.
This option may seem good, but there is the problem of a random point $A$ forming a line segment of odd length with point $B$ and another line segment of even length with point $C$. This idea could apply to other points in the set which seemingly only create line segments of even length.
$(3)$ Exclude all points which form line segments of odd length with other points.
This option poses the same problem as option $(2)$ where a random point could also form a line segment of even length with another point.
I have not thought of any other options to fix this problem. Can someone help me out?