Simple Application of Model Theory
Rimu Shuang. "Untitled Photo". Nov 24, 2013. Under a Creative Commons 3.0 Attribution License.
November 25, 2013
Recently I’ve been taking a class in model theory and my undergraduate thesis will probably touch on model theory in some way. As such I’ve decided to write a little about why anybody should care about model theory.
First off, model theory is a branch of mathematics that straddles the boundary between logic and math. Most model theory deals with first order logic and what kind of mathematical structures satisfy certain first-order statements. In essence model theory is built around the idea of differentiating what can be said about a mathematical structure from the mathematical structure itself. Of course we usually modify “what can be said” to say “what can be said within first order logic” because the philosophical question arises of whether mathematical structures exist independently of whether we are able to describe them. Although model theory does crop up in philosophy, for now we can avoid this (which is too philosophical and mystical for my taste) path of inquiry.
Now, even for most other branches of math, studying the very logic systems that gird mathematics seems a bit of overkill. After all, questions of the merits of first-order logic, the restrictions that such logics impose on us, and the study of general mathematical structures seems too abstract even for much of mathematics.
Instead to show the power of model theory, it is useful to consider an example which our professor showed us on the first day of our class and which will be the topic of this post (note that as a result I did not come up with this example). Euclid, the famous Greek mathematician, had a series of geometric postulates as follows:
Between any two points there can be drawn a line.
Any finite line segment can be extended infinitely in both directions.
Any circle can be described by a center and a radius.
All right angles are equal to each other.
(Parallel Postulate) Given any line L and point not contained in the line, there exists exactly one line which is parallel to L. Note that this is not Euclid’s original axiom, but it turns out to be logically equivalent.
Following the spirit of Euclid and the conditions that the mathematicians investigating these postulates were under, I won’t attempt to provide very rigorous definitions of what these mean. Instead I’ll just provide a short note to say that two lines are parallel if the minimum distance from one point p of one line to the points on the other line remains constant for any choice of p.
For several millennia, it seemed that Euclid’s parallel postulate should be provable from the other four, mainly based off an intuition that the fifth postulate was more “complicated” than the others, especially since the other four seemed more to be definitions and the parallel postulate seemed to be more of a theorem. Some people might be thinking right now, “ah ha hyperbolic geometry is the counter-example!” And they would be right, but it turns out that the thought process behind proving the independence of the parallel postulate is an archetypical example of the thinking that drives model theory.
Model theory concerns itself first with “theories,” sets of “sentences” which are in turn, roughly, syntactically valid collections of symbols drawn from some agreed upon language. The second object that model theory concerns itself with is the mathematical structures which fit within the constraints of some given theory. Such a structure is said to “satisfy” the associated theory. So for example if I have a theory that consists of the single sentence “This structure must have an element,” then any mathematical structure except the empty set satisfies this theory. Likewise, if I have a theory that consists of the group axioms, then any group satisfies this theory.
In this way model theory sort of turns traditional mathematical theories on top of their heads. Normally we think of a mathematical theory such as field theory to define a set of mathematical objects which we then call “fields.” Model theory, on the other hand, presupposes the existence of fields and says that instead it just so happens that they satisfy a theory that we then call “field theory” (well it is possible to take a non-Platonist view, but this is generally the easiest).
Thus, while traditionally it doesn’t make too much sense to ask what mathematical objects satisfy, say, the axioms of set theory, e.g. ZFC, since we usually specify the theory first and then say everything defined by this is a “set.” It turns out that there are many different models of set theory (i.e. there are different notions of exactly what constitutes the universe of sets, even under ZFC). I’ll save that discussion for another time though.
Turning back to our theory at hand, namely the Euclidean postulates, if we are able to find a mathematical structure which satisfies the first four postulates, but does not satisfy the parallel postulate, then we know that the parallel postulate must be logically independent of the other postulates, i.e. the other postulates cannot prove the parallel postulate. If they could prove the parallel postulate, then every mathematical structure we could find that satisfied the first four postulates must then also satisfy the parallel postulate.
And that is where hyperbolic geometry comes in. In fact, however, there is an even simpler example of a mathematical structure which satisfies the first four postulates, but does not satisfy the last postulate. We can use the sphere and define lines as great circles on our sphere and points as points on the surface of the sphere.
We can verify each of the postulates one by one.
Any two points on the surface of a sphere can be connected by a great circle.
Any arc of a great circle can be extended into an entire great circle (let’s call the length of a great circle infinite).
Circles can be defined as projections of a circle in a plane on the spherical surface.
Right angles are indeed all equal.
The parallel postulate does not hold because given any two great circles, they must intersect at some point.
Hence via a simple application of model theory, we’re able to solve a problem that plagued mathematicians for thousands of years. Yay model theory!