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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.

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    $\begingroup$ If geometry has a language, axioms and it uses any "reasonable" logic as a backbone, then you can interpret it just like you can interpret the real numbers. I'm not sure that I understand the actual difficulty. $\endgroup$ – Asaf Karagila Jun 7 '14 at 7:04
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    $\begingroup$ You don't have to work very hard to develop the basic geometry on the plane. After you've defined $\Bbb R$, you consider subsets and points in $\Bbb{R\times R}$ as your lines and points. It seems to me that you don't really understand how formalizing things into set theory works. Perhaps it would benefit you to learn some basic set theory and logic first. $\endgroup$ – Asaf Karagila Jun 7 '14 at 11:44
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    $\begingroup$ "I thought ZFC was the foundation for most of modern mathematics?" Sigh ... many people still believe and advocate this old point of view. ZFC has absolutely nothing to do with geometry and, by the way, cannot really explain what is going on in algebra, topology and other related branches of mathematics. Of course it is possible to model all these theories in ZFC. But it doesn't mean that ZFC and therefore set theory is the most appropriate foundation for mathematics. If you want to understand the foundations and general principles of mathematics, you have to learn category theory. $\endgroup$ – Martin Brandenburg Jun 7 '14 at 13:16
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    $\begingroup$ @Martin: I think I have a reasonable basic handle on foundations of mathematics, but I only took one course that dealt with categories. So "have to"? I hardly think so. Of course, if you believe that mathematics should be founded on categories then it is advisable to learn category theory. But you don't have to believe that, just like you can sigh with exasperation whenever someone says that mathematics is what is done inside set theory. $\endgroup$ – Asaf Karagila Jun 7 '14 at 13:36
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    $\begingroup$ @ignoramus, exists a set-theoretic version of Hilbert’s axioms: citeseerx.ist.psu.edu/viewdoc/… $\endgroup$ – mle Jun 8 '14 at 17:04
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Basically, to develop "formally" a geomety you have two ways; call them analytic and synthetic respectively.

Analytic

This is our "good old" Analytic geometry :

a point in the space is a ordered triple of real numbers : $(x_1,x_2,x_3)$

a line is the totality of points $(x_1,x_2,x_3)$ such that $u_1x_1 + u_2x_2 + u_3x_3 = 0$, where at least one $u_j (j = 1,2,3)$ is different from zero

and so on ...

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 :

space will be regarded as a set $S$; the points of space will be the elements of this set. We will also have given a collection of subsets of $S$, called lines, and another collection of subsets of $S$, called planes.

Thus the structure that we start with is a triplet : $<\mathcal S, \mathcal L, \Pi>$, where the elements of $\mathcal S, \mathcal L, \Pi$, and are called points, lines and planes, respectively.

Our postulates are going to be stated in terms of the sets $\mathcal S, \mathcal L$, and $\Pi$.

Here are the first two postulates :

I-0 : All lines and planes are sets of points.

I-1 : Given any two different points, there is exactly one line containing them [we can "trivially" express the fact that the point $Q$ is contained into the line $l$ with the formula : $Q \in \mathcal S \land l \in \mathcal L \rightarrow Q \in l$ ].

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.

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  • $\begingroup$ With you except for maybe the last line. From what you said in the rest of your answer, it would seem that set theory is indeed the foundation of most of mathematics -- starting, of course, from different initial assumptions. $\endgroup$ – Dan Christensen Jun 8 '14 at 4:00
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    $\begingroup$ @DanChristensen - "foundation" is a loaded term... For sure set language is a sort of esperanto for math, a common (and simple) language able to express (quite) all math facts. About the "traditional" foundationals point of view of last century (Frege/Russel, Hilbert, Brouwer) to sat that set theory has dissiped the issue regarding "foundations" (i.e.certainty, evidence, etc.) it is hard to say. $\endgroup$ – Mauro ALLEGRANZA Jun 8 '14 at 13:02

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