Faculty Projects: Axiomatic Set Theory with Mr. Letarte

Commonwealth School teachers bring an infectious intellectual energy to their classrooms, fueled, in part, by their own innate curiosity. What happens when that curiosity is unleashed? The Hughes/Wharton Fund for Teachers aims to do just that. Named after the late John Hughes, who taught English at Commonwealth for nearly thirty years, and after recently retired Head of School Bill Wharton, who founded the original Hughes fund in 2011 and championed faculty scholarship throughout his tenure, the Hughes/Wharton Fund ensures faculty can pursue their academic passions, access fulfilling professional development opportunities, and have the latitude to create new courses and reinvigorate existing ones. 

Like their students, Commonwealth teachers love language. And for math teacher Al Letarte, poring over “the beautifully concise formal language” that animates Axiomatic Set Theory made for enormous summer fun. (Who needs the beach?) Through Commonwealth’s new Axiomatic Set Theory course, Mr. Letarte gives students a chance to explore concepts usually reserved for college math majors, from the syntax and semantics of first-order languages to the very nature of mathematical truth.

I had great fun this summer figuring out how to design a rigorous introduction to axiomatic set theory at Commonwealth. The course I came up with should work well in our curriculum in the sense that its content is “entry-level,” allowing strong students to jump in with no prior track record in our advanced mathematics electives. That said, I expect that it will capture and hold the attention of even the best of our students throughout the school year. 

The period from the 1830s to the 1880s saw remarkable proofs by modern methods of the unsolvability of a number of very old problems in geometry: trisection of the angle (Wantzel, 1836), doubling of the cube (Wantzel, 1837), proof of Euclid’s fifth postulate (Beltrami, 1868), and quadrature of the circle (Lindemann/Weierstrass, 1882). These breakthroughs—particularly the proof of the independence of the parallel postulate, since it is a relative consistency result—provided inspiration for the development of modern mathematical logic, of which axiomatic set theory is a part. (Note to the lay reader: relative consistency of an assertion means that the assertion can hold true in the presence of specified axioms. If both the assertion and its negation are consistent with a system of axioms, it is said to be independent of the axioms.) 

The course is devoted to a study of ZF, a formal system due to Zermelo and Fraenkel in which all of mathematics can be constructed from a surprisingly simple set of axiomatic assertions, expressible in a beautifully concise formal language, about a collection of primitive objects known as “sets.” ZF and just one additional assertion, the axiom of choice, allow us to build a hierarchy of set objects rich enough to capture all of the mathematics that anyone but a small group of logicians would ever need to do. 

Each model of ZFC contains the usual mathematical objects, such as natural numbers, that behave as expected. Surprisingly, the axioms of set theory allow us to extend the natural numbers to a hierarchy of sets known as ordinals that serve as “templates” for an important type of mathematical structure known as a well-ordering. The hierarchy of ordinals, while itself a well-ordering, is too large to be a set. Special ordinals known as cardinals act as “measuring sticks” for assessing the relative sizes of sets (the notion of relative size, though simple and intuitive for finite sets, turns out to be mind-bogglingly complicated for infinite ones). Transfinite induction and recursion, generalizations of techniques of the same name on the set of natural numbers, illuminate the workings of ordinal and cardinal arithmetic, transporting us quite literally to infinity and beyond. Time permitting at the end of the course, we will explore some famous assertions such as the continuum hypothesis that mark the boundaries of what can be definitively known in a model of set theory. 

Set theory and model theory (the approach I normally use when teaching mathematical logic to beginners) are different entry points into the field. Either of these requires brushing aside references to the other whenever they arise. For instance, the model-theoretic presentation I normally use makes frequent use of cardinal arithmetic, which students in a first course in model theory will not have studied. In set theory, on the other hand, I am having to provide students with a crash course on the syntax and semantics of first-order languages—a topic that would be addressed more thoroughly in a course on model theory.

Much of our time in class during the first weeks of the course was devoted to philosophical discussions about the nature of mathematical truth and to an informal description of the hierarchy of set objects we intend to build within our axiomatic framework. But we have now given a precise description of syntax and semantics in the language of set theory (itself a mathematical object!), and are beginning to lay out the axioms themselves. 

Needless to say, having the opportunity to introduce bright, curious high school students to a topic as cutting-edge as this one is as rewarding for me as I expect it will be for my audience.

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