Sets

Broadly speaking, a set is a collection of objects, and in mathematical discourse these objects are mathematical ones such as numbers, points in space or other sets. If we wish to rewrite sentence(3) symbolically, another way to do it is to define P to be the collection, or set, of all prime numbers.

Then (3) can be rewritten, “5 belongs to the set P”.

This notion of belonging to a set is sufficiently basic to deserve its own symbol, and the symbol used is 2. So a fully symbolic way of writing the sentence is 5 2 P.

The members of a set are usually called its elements, and the symbol 2 is usually read “is an element of”. So the “is” of sentence (3) is more like 2 than =. Although one cannot directly substitute the phrase “is an element of” for “is”, one can do so if one is prepared to modify the rest of the sentence a little.

There are three common ways to denote a specific set. One is to list its elements inside curly brackets: {2, 3, 5, 7, 11, 13, 17, 19}, for example, is the set whose elements are the eight numbers 2, 3, 5, 7, 11, 13, 17 and 19.

The majority of sets considered by mathematicians are too large for this to be feasible - indeed, they are often infinite - so a second way to denote sets is to use dots to imply a list that is too long to write down: for example, the expressions {1, 2, 3, . . . , 100} and {2, 4, 6, 8, . . . } represent the set of all positive integers up to 100 and the set of all positive even numbers respectively.

A third way, and the way that is most important, is to define a set via a property: an example that shows how this is done is the expression {x : x is prime and x < 20}. To read an expression such as this, one first says, “The set of”, because of the curly brackets. Next, one reads the symbol that occurs before the colon. The colon itself one reads as “such that”. Finally, one reads
what comes after the colon, which is the property that determines the elements of the set. In this instance, we end up saying, “The set of x such that x is prime and x is less than 20,” which is in fact equal to the set {2, 3, 5, 7, 11, 13, 17, 19} considered earlier.

Many sentences of mathematics can be rewritten in set-theoretic terms. For example, sentence (2) earlier could be written as 5 2 {n : n < 10}. Often there is no point in doing this - as here where it is much easier to write 5 < 10 - but there are circumstances where it becomes extremely convenient.

For example, one of the great advances in mathematics was the use of Cartesian coordinates to translate geometry into algebra, [Cross references, such as Kollar and my remarks on geometry later in this section.] and the way this was done was to define geometrical objects as sets of points, where points were themselves defined as pairs or triples of numbers.

So, for example, the set {(x, y) : x2 + y2 = 1} is (or represents) a circle of radius 1 about the origin (0, 0). That is because, by Pythagoras’s theorem [CR?], the distance from (0, 0) to (x, y) is p x2 + y2, so the sentence “x2 + y2 = 1” can be reexpressed geometrically as “the distance from (0, 0) to (x, y) is 1”.

if all we ever cared about was which points were in the circle, then we could make do with sentences such as “x2 + y2 = 1”, but in geometry one commonly wants to consider the entire circle as a single object (rather than as a multiplicity of points, or as a property that points might have), and then set-theoretic language is indispensable.

A second circumstance where it is hard to do without sets is when one is defining new mathematical objects - unless they are exceedingly simple. Very often such an object is a set together with a mathematical structure imposed on it, which takes the form of certain relationships amongst the elements of the set. For examples of this use of settheoretic
language, see the later sections on number systems and algebraic structures. Sets are also very useful if one is trying to do metamathematics, that is, to prove statements not about mathematical objects but about the process of mathematical reasoning itself.

For this it helps a lot if one can devise a very simple language - with a small vocabulary and an uncomplicated grammar - into which it is in principle possible to translate all mathematical arguments. Sets allow one to reduce greatly the number of parts of speech that one needs, turning almost all of them into nouns. For example, with the help of the membership symbol 2 one can do without adjectives, as the translation of “5 is a prime number” (where “prime” functions as an adjective) into “5 2 P”has already suggested.

This is of course an artificial process - imagine replacing “roses are red” by “roses belong to the set R” - but in this context it is not important for the formal language to be natural and easy to understand. [Cross-reference to Ellenberg’s article in Section IV for further discussion of adjectives.]