Reference request for set-theoretic foundations of geometry

Question

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?

Answer 1

See :

• Gerard Venema, Foundations of Geometry (2011), Appendix B : Systems of Axioms for Geometry, page 350-on

and :

• William Richter, A Minimal Version of Hilbert’s Axioms for Plane Geometry.

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.

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See also :

• Francis Borceux, An Axiomatic Approach to Geometry : Geometric Trilogy I (2014), Ch.8 Hilbert’s Axiomatization of the Plane, page 305-on.

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For axiomatizations, see :

• Hilbert's axioms (David Hilbert, Grundlagen der Geometrie (1899, Engl.tr.The Foundations of Geometry)

• Oswald Veblen's axioms, (Oswald Veblen, A System of Axioms for Geometry, Transaction of the AMS, 1904)

• Birkhoff's axioms, build upon the real numbers (George David Birkhoff, A Set of Postulates for Plane Geometry (Based on Scale and Protractors), Annals of Mathematics 1932)

• Tarski's axioms : formulated in first-order logic with identity, and requiring no set theory (Alfred Tarski, What is elementary geometry?, in Leon Henkin, Patrick Suppes and Alfred Tarski (editors), The axiomatic method. With special reference to geometry and physics. Proceedings of an International Symposium held at the Univ.of Calif., Berkeley, 1958, Studies in Logic and the Foundations of Mathematics, 1959)

Answer 2

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.

Answer 3

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.