What is a set? (revisited)

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’.

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