I've been trying to get a rigorous understanding for the mathematical concepts I learned in high school. I've been reading about how the real numbers can be constructed from the axioms of ZFC, but I can't find any information on geometry.
I've read about axiomatic formulations of Euclidean geometry, such as Hilbert's, but they seem to introduce new primitive notions.
I've read definitions of Euclidean spaces as sets, but I don't see how it's isomorphic to the synthetic geometry taught in high school. I'd like to be able to know the answers to questions such as:
- How is the concept of "angle" defined?
I thought ZFC was the foundation for most of modern mathematics? Doesn't this mean that it should be possible to describe basic high school geometry topics (circle geometry, cartesian geometry, etc) using ZFC?
My goal is to know that synthetic geometry is isomorphic to something analytically defined using ZFC, so I can know that any proofs in synthetic geometry will also hold in ZFC.
Sorry if this is impossible. I am very ignorant of how mathematics works.
Best Answer
Basically, to develop "formally" a geomety you have two ways; call them analytic and synthetic respectively.
Analytic
This is our "good old" Analytic geometry :
But real numbers are definable in set theory; thus - in principle - you can translate into set-theoretic notation the equation of the line.
Synthetic
See Edwin Moise, Elementary Geometry from an Advanced Standpoint (3rd ed - 1990), page 43 :
Here are the first two postulates :
We write $\overline{PQ}$ for the unique line containing $P$ and $Q$.
We define the relation of betweenness between (sic !) three points $P, Q, R$.
Then [see pages 64-65] : if $R,Q$ are two points, the segment between $R$ and $Q$ is the set whose points are $R$ and $Q$, together with all points between $R$ and $Q$.
The ray $\overrightarrow {AB}$ is the set of all points $C$ of the line $\overline {AB}$ such that $A$ is not between $C$ and $B$. The point $A$ is called the end point of the ray $AB$.
An angle is the union of two rays which have the same end point, but do not lie on the same line. If the angle is the union of $\overrightarrow {AB}$ and $\overrightarrow {AC}$, then these rays are called the sides of the angle; the [common] end point $A$ is called the vertex.
Finally, you can "close the circle" between this two approaches.
Assuming that we have defined the set $\mathbb N$ of natural numbers inside set theory [but I prefer to say that we have defined a model of the natural number system], and then the set $\mathbb R$ of real numbers, we can use $\mathbb R^3$ and call it : (three-dimensional) space.
Comment
What have we gained so far ? I think nothing more and nothing less than what we already have with Descartes' discovery of analytic geometry : an "embedding" of the euclidean geometry into the "cartesian plane".
Of course, the "basic" set-theoretic language gives us a powerful tool for expressing also geometrical "facts" : we can write $P \in l$ for : "the point $P$ is contained into line $l$", we can write $l_1 \cap l_2 \ne \emptyset$ for "two lines intersect each other", ...
But I think that speaking of "foundation for most of modern mathematics" can be mesleading.