Posted in May 2019

This is an addendum to the first in a short series of blog posts intended to explain the terms in red in the following sentence, that succinctly describes the Continuum Hypothesis.
There is no set whose cardinality is strictly between that of the integers and the real numbers.
These are, in the words of John Lloyd, the bits that he does not understand.

The gist of the post referred to above was that, very strictly speaking, sets have no proper mathematical definition. The definitions that were quoted from the works of Bolzano and Cantor were, to a large extent, by synonym: “a set is a collection …”. The ellipsis would contain some conditions what the collection should satisfy to be deemed a set. But the definition would be incomplete because `collection’ remained undefined. Many definitions in mathematics suffer from a similar `defect’: at some point there is a primitive notion that is not further defined. In most every branch of mathematics that primitive notion turns out to be `set’ in some form or another.

That looks bad: Mathematics seems to be based on a badly defined notion. However not all is lost. Most of the time we know exactly what we are talking about. It is true that `collection’ is undefined but, as mentioned in the original post, we recognize one when we see one and in Mathematics we are very particular about how we work with them.

By way of example consider the books currently in our house. They form a well-defined collection: it is very clear which books are in that collection and which books are not. That collection forms what Bolzano and Cantor consider to be a set. It has an unambiguous definition. Just like the set of books in our house that have exactly 250 pages: everyone that we to gather `the books with exactly 250 pages’ will come back with the same collection. And the unambiguity separates the sets from the arbitrary collections. If I were to ask John Lloyd to gather the interesting books in our house he would most likely come out with a different collection than I would. The phrase `the interesting books in our house’ does not define a set.

What determines a set in mathematics is the unambigiuty of its definition: no matter who we set the task of determining what is in it, the answer should always be the same. That does not mean that that task is easy or doable in a (very) short time. The prime numbers form a set, a subset of the set of natural numbers, and for every individual natural number it is straightforward to determine whether it is prime or not, but separating them from the other natural numbers by hand is not an option.

Because of this and other examplese have developed the curly-braces notation for sets.
P = {n : n is a prime number}
is a properly defined set and for every natural number n we can decide whether n∈P (`n is in P’) or not.

And that is how mathematicians consider sets: as collections where membership can be checked unambiguously. Thus the advisoy committee of Episode 2, series 10 of the Museum of Curiosity forms a set, the funny members of that committee most likely do not. There is, alas, no unambiguous definition of `funny’.

This is the first in a short series of blog posts intended to explain the terms in red in the following sentence, that succinctly describes the Continuum Hypothesis.
There is no set whose cardinality is strictly between that of the integers and the real numbers.
These are, in the words of John Lloyd, the bits that he does not understand.

Sets pervade mathematics. Basically every definition of a mathematical structure contains the phrase “… a set such that …”. The language and tools of Set Theory generally make it possible to formulate results efficiently.

It may therefore come as a bit of a surprise that the question “What is a set?” does not have a straightforward answer. That sounds strange because we generally recognise a set when we see one: thimbles, forks, railway-shares …,(but not care, hope and soap) you name it, someone will have a set (collection) of it.

However, in Mathematics we like precise definitions, so that at every moment it is clear what we are talking about. A word of warning is needed here, nicely illustrated by this quote from Goethe.

1005. Die Mathematiker sind eine Art Franzosen; redet man mit ihnen, so übersetzen sie es in ihre Sprache, und dann ist es alsobald ganz etwas anderes.
Johann Wolfgang von Goethe, Maximen und Reflexionen, Nachlass, Über Natur und Naturwissenschaft

I do not have the illusion to know what Goethe actually meant to say with this and further study of his work may reveal that, but for me this quote is very apt all by itself. Many definitions of mathematical notions do not conform to the expectations of non-mathematicians. Things that are nigh on synonymous in the dictionary may have rather different meanings in mathematics.

To define what a set is we turn to two pioneers of the study of the infinite, Bernard Bolzano and Georg Cantor.

In his Paradoxieen des Unendlichen Bolzano wrote this on page 4, after a short introduction wherein he exlained the need for a definition of Menge.

Einen Inbegriff, den wir einem solchen Begriffe unterstellen, bei dem die Anordnung seiner Teile gleichgültig ist (an dem sich also nichts f&uumlr uns Wesentliches ändert, wenn sich bloß diese ändert), nenne ich eine Menge;

In the translation of this work by Donald A. Steele and the above definition is rendered as follows.

An aggregate whose basic conception renders the arrangement of its members a matter of indifference, and whose permutation therefore produces no essential change from the current point of view, I shall call a set (Menge),

The very first words written by Georg Cantor in his Beiträge zur Begründung der transfiniten Mengenlehre are

Unter einer ,Menge` verstehen wir jede Zusammenfassung M von bestimmten wohlunterschiedenen Objecten m unserer Anschauung oder unseres Denkens (welche die ,Elemente` von M genannt werden) zu einem Ganzen.
In Zeichen drücken wir dies so aus:
M={m}

In the translation by Philip E. B. Jourdain we find:

By an “aggregate” (Menge) we are to understand any collection into a whole (Zusammenfassung zu einem Ganzen) M of definite and separate objects m of our intuition or our thought. These objects are called the “elements” of M.
In signs we express this thus:
M={m}

I think it is no coincidence that the notion `Menge’ was defined during investigations of the notion of `infinite’. At that moment the relations between the individuals that make up the whole are of secondary importance. And at some point one chooses one out of many synonyms &mdsah; collection, multitude, Mannigfaltigkeit, aggregate, set, verzameling, &hellip — and that becomes that name of the basic object of investigation.

If you read the definitions closely then you will see that they, strictly speaking, define nothing: both use a synonym, Inbegriff or Zusammenfassung, as a definition. However, Bolzano explicitly adds a condition (and Cantor does so implicitly, as witnessed by the rest of his paper) that was also mentioned above: the relations, if any, between the elements of the sets are not important. A few paragraphs before the definition Bolzano used the example of a broken tumbler; we tend to view that as different from that same tumbler before it was broken because the relations between the constituents have changed, as a set — of atoms, molecules — it has not changed.

It is this condition that tell us what the goal of Bolzano and Cantor undoubtedly was: delineate as sharply as possible about which objects they make their pronouncements. Bolzano’s definition was the summary of a long run-up where he discussed what properties a Menge should have. Cantor jumped right in because he had been considering sets for two decades already.

On a naïve level these definitions are quite workable because all that happens is that certain entities now have the label `set’ applied to them. Something like {1,2,3,4,5} is recognised by everyone as a “the set of natural numbers from 1 through 5”. And also sets with a description like {n ∈ N : n ≤ 10<sup>100</sup>} is fine, provided we have learned some of the language of mathematics. Here ∈ means `is element of’, ≤ means `less than or equal’ and N represents the set of natural numbers.

Mathematics differs from `daily life’ in one seemingly innocuous point: mathematicians give set status to a few things that some people do not recognise as sets. The empty set and sets with (exactly) one element are perfectly acceptable mathematically. But, if I were to tell you that I have a set of stamps and show you an album without any in it you would not consider me a stamp collector, nor if a were to show you just one stamp (right before I stick it on an envelope).

Mathematically speaking these are legitimate sets and they are also quite necessary because it would become quite cumbersome to exclude them as results of operations on sets. Think of equations. Very often we speak of solution sets and that would suddenly be illegal is the equation had no or just one solution? Really?

But still, this all assumes that we recognize a set when we see one: a collection of things thrown together between curly braces for a certain purpose. It’s the thirteen cards in your hand at bridge before you inspect and order them; it’s the points on a line where it is immaterial which point lies to the left or right of another point. All this does not tell us what a set is. For that we should define first what a collection is …

So, where does that leave us? The tools and language of Set Theory pervade mathematics and are quite powerful, yet we do not have a fully satisfactory definition of what a set actually is. For day-to-day mathematics that is no big problem because, as I said above, we recognise many familiar entities as `sets’ and treat them as such.

But what about those of us who want to know what a set really is? Who do not want to `recognise a set when they see it’? Well, we can satisfy them by setting up Set Theory purely logically and thus define what our objects are. The resulting `sets’ are not quite like those we learned to recognise but every one of our familiar sets has a faithful logical copy. This means that we can, with a bit of care, keep on using sets in the naïve way we have always done.

We may come back to that logical approach in a later post.