Infinite reactions, 3

Here is a third posting on reactions to “To Infinity and Beyond” – a bit different from the others.

The theorem in Euclid stating that the base angles of an isosceles triangle are equal is referred to as the Pons Asinorum, or “Bridge of Asses”, for two reasons. Firstly, the diagram accompanying the theorem looks a bit like a bridge. Second, because asses cannot cross the bridge; that is, if you can’t understand this theorem you will never be a mathematician.

I claim that the modern Pons Asinorum is the Equivalence Relation Theorem. This states, informally, that an equivalence relation (a reflexive, symmetric and transitive binary relation on a set) does exactly the same job as a partition of the set (a collection of non-empty subsets covering the set without overlap). Given any equivalence relation, the set of equivalence classes is a partition; given any partition, there is a unique equivalence relation whose equivalence classes are the parts of this partition.

The theorem is not so difficult in itself. The reason why I feel that the name is appropriate is the way in which it is used. In algebra, we define quotient groups, rings, etc. to be objects whose elements are the equivalence classes of an equivalence relation on another set. Students find this level of abstraction quite testing. For example, the ring Z₄ of integers mod 4 is a structure with four elements, each element being a congruence class mod 4 (an infinite set). Thus the first element is the set {…,–4,0,4,8,12,…}, and so on. Not surprisingly, many students (and others) take refuge in the view that Z₄ is a ring whose four elements are {0,1,2,3}, the operations being addition (or multiplication) followed by taking the remainder on division by 4. Actually this is much less satisfactory; in this case there is such a convenient choice of representatives of the classes, but in general there is not.

The point is that here is a finite set (with just four elements), whose elements are themselves infinite sets.

Anyway, this morning, after trying to persuade my lecture class to get their heads round this, I returned to my office and read my email. There was one from a correspondent about issues raised by the Horizon programme. His confusion seemed to be that, because the set of natural numbers is infinite, a natural number is an infinite object; he had no problem with the infinite natural number …3,333,333 which could be matched up with the real number 1/3 in a bijection between the natural numbers and the real numbers.

So the natural numbers present the opposite difficulty to Z₄; an infinite set whose elements are finite.