# Reference request for set-theoretic foundations of geometry

My question is,

Is it possible to define geometrical concepts (say, of Euclidean Geometry) like 'point', 'striaght line' in purely set theoretic terms?

So far, I could think of the following definitions,

Let $S$ be a non-empty set.

• An element of $S$ is said to be a point.

• $S$ will be said to be a plane if there exists two sets $X$ and $Y$ such that, $$S=X\times Y$$

• $S$ will be said to be a space if there exists three sets $X$, $Y$ and $Z$ such that, $$S=X\times Y\times Z$$

But the problem is that I couldn't define a straight line in this scheme. This made me wonder if there is any foundation of geometry which is based only on set theory?

If so, then can some related literature be mentioned?

See :

and :

For plane geometry :

We are given a set which is called a plane. Elements of this set are called points. We are given certain subsets of a plane called lines. [...] we discuss one plane, referred to as `the plane'.

A line $l$ is then the set of all point $P$ in the plane so that $P \in l$. Two lines $l$ and $m$ are then equal if $P \in l$ iff $P \in m$, for all points $P$.

We have as usual the intersection $l \cap m$ of two lines $l$ and $m$, the set of points $P$ in the plane such that $P \in l$ and $P \in m$. We will call points collinear if there is a line in the plane containing them.

For axiomatizations, see :

• It seems to me that there is slight difference in the goal of Richter's paper and what I want. To be precise, in Richter's paper he says that we are given 'certain subsets' of a plane called line, whereas he doesn't specify exactly what type of subsets should we call lines. My requirement was to base everything on the sole mathematical object point and then construct lines, planes and so on. – user 170039 Dec 24 '15 at 5:21
• @user170039 The points themselves don't contain this information. You need points along with "certain subsets" of the points which will be defined as the lines. The rest (planes, etc) can follow from here. – Morgan Rodgers Jan 6 '16 at 12:26

I don't really understand your definition of a plane; perhaps by "set" you mean "line"? In any event, you can implement these kinds of definitions using lattice theory. We start off with an atomistic joinlattice $J$. Imagine that $J$ is the lattice of affine subsets of a finite-dimensional affine space. We then define that:

• a point in $J$ is an atom of $J$
• a line in $J$ is an element of $J$ that can be expressed as a join of two points, but no fewer.
• a plane in $J$ is an element of $J$ that can be expressed as a join of three points, but no fewer.
• etc.

More generally, we define that a $k$-flat of $J$ is an element of $J$ that can be expressed as a join of $(k+1)$-many elements, but not of $k$-many elements. To make sure your "abstract geometries" behave correctly, you can then focus your attention on atomistic lattices in which for all $k$, the $k$-flats form an antichain. There's also a notion that I refer to as well-rankedness that's quite relevant.

Anyway, this may or may not be what you're looking for. I offer several references:

1. If you're willing to use the concept of a vector space to help formalize Euclidean geometry, then I highly recommend Audin's Geometry. This is by far the best book on Euclidean geometry that I've ever seen.

2. For a more "set-theoretic" perspective, Coppel's Foundations of convex geometry is brilliant, insofar as it manages to establish Euclidean geometry without recourse to vector spaces. The book Join Geometries also looks pretty good, although I've never read it.

3. If you're interested in perspective 2, you should probably run off and learn some lattice theory (because the convex subsets of any real affine space form a lattice, and the join operation has enormous geometric significance). Davey and Priestley's Introduction to Lattices and Order is a nice introduction.

4. If you're interested in differential equations or theoretical physics, you'll probably want to learn geometry the "differential geometry" way, which involves differentiable manifolds and Riemannian manifolds. I don't have any particular books to recommend here; I have yet to find one that is to my taste.

5. There's also something called "metric geometry" that takes metric spaces as its foundation. I don't know much about it, but you could try googling for more information.

Intermediacy is one of the most fundamental concepts of geometry. The properties of intermediacy are nailed down by certain non-set theoretical axioms. For example (in Euclidean and Hyperboloc geometry):

• If $A$ is between $B$ and $C$ then $C$ is not between $A$ and $B$.
• If $A$ is between $B$ and $C$ then there exists a $D$ such that $C$ is between $A$ and $D$.

Where the word between is not defined but by the axioms above and by a few other axioms.

The concept of the straight line can be defined then in the Veblen axiomatization of geometry. The concept of the point and the concept of betweenness remains undefined. The fact that the objects in geometry are elements of a set does not make geometry a subdiscipline of set theory.

This is very common in mathematics. Set theory provides a framework within which specific axioms characterize a specific discipline like Euclidean geometry.

The objects in your definitions are not well defined.

Consider, e.g., $X=Y=X={1}$. Then a point, a plane and a space have one element. The other example: Cartesian product of two circles is a torus. One can call it a plane, but it is rather strange, isn't it?

• I don't understand what you wanted to mean by your first example. Can you clarify a bit? Also I don't understand how you treat $1$ as a point. Shouldn't it be the singleton set $\{\emptyset\}$. In that case $\emptyset$ will be called a point, not $1$. It seems to me that (but I may be wrong, of course) what you are objecting is the fact that if both a point, a plane and a space have one element then there will be some confusion regarding the term 'point'. But in your example, the plane is $\{(\emptyset,\emptyset)\}$ and the element can be called a point in this case too. ... – user 170039 Dec 24 '15 at 3:42
• ...But in this case it will be called, if you want more precision, a point in the plane. Similar will be the case for space. – user 170039 Dec 24 '15 at 3:44