# How can be set -theoretically defined the whole collection of possible circles in a given plane P ? ( not using analytic geometry)

[ edited 9th april 2019]

I'd like to solve this question using only basic concepts of geometry ( without analytic geometry) and of elementary set theory.

If I am correct, the following process allows me to " build" the desired set. But, how could I express, in a single formula,the result of this process, using the proper logical and set theoretical symbolism?

(1) I choose an arbitrary point O (as first "center").

(2) I define an equivalence relation : the set (of points) A is equivalent to the set (of points) B iff all the elements of A are at the same distance from O as are all the elements of B ; I obtain an infinity of equivalence classes ( one for each possible " orbit")

(3) inside each equivalence class ( that is , for each " orbit") I choose the greatest set using the inclusion relation ( for the circle is the greatest set amongst sets whose points are all at a given distance from a given "center"); I obtain an infinity of circles for the first center O.

(4) I repeat (1)-(3) for each point in the plane, that is, for each possible "center".

(5) Using, maybe, the union operation, I gather all my ( infinite) collections of circles for a given center in a new set , which would be the set of circles in a plane P.

Remark - This is not homework; it is a question I ask myself, as a gratuitous exercise in logic/ elementary set theory

• Pay attention to the difference between a set and its points. The set CIRCLE you defined is a set of subsets. The union of all subset is all the plane (every point is contained in some circle) but the set CIRCLE is not the plane Apr 8, 2019 at 13:56
• Why not $C=\{(x,y,r): r\gt 0\}$? Apr 8, 2019 at 13:59
• @ John Douma - Is a triple sufficient? Would we need a quadruple in order to take the center into account ? Or not?
– user654868
Apr 8, 2019 at 14:05
• @Eleonore Saint James - I think (x,y) is the centre.
– Paul
Apr 8, 2019 at 14:08
• I think I understand in which way a given circle can be identified by a triple < x, y, r> x and y being the coordinates of the center and r the radius. But I am wandering whether this triple is " able" to define the circle as a set of points. Is it the case?
– user654868
Apr 8, 2019 at 14:16

Mathematicians already have a symbol for a circle of radius $$r$$ centred at $$(x_0 ,y_0 )$$, it is $$B((x_0 ,y_0 ),r)=\{ (x,y) | (x-x_0 )^2 + (y - y_0 )^2 = r \}$$ Then, the set of all circles is $$C=\{B((x_0 ,y_0 ),r) | r>0, (x_0 ,y_0 ) \in Plane \}$$
• You could let $\vec{x}$ denote a point on the plane, then as mentioned in previous comments, every open ball $B(\vec{x}, r)$ can be "labelled" by an ordered pair $(\vec{x},r)$. And the set $\{(\vec{x},r) |r>0 \}$ is isomorphic to the set of all circles, since every circle has a label, and every label corresponds to a circle (it is a bijection). Is that what you are looking for? Apr 8, 2019 at 15:34