Strictly between

This is the third 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.

Thus far we have dealt with set in this post and this one, and with the notion of cardinality in this one.

To recap our findings: we first came to the disappointing realisation that the definitions proposed by Georg Cantor, very strictly speaking, did not deliver on their promises. There is a way out of this via the development of Axiomatic Set Theory but that would take us too far afield.
In the spirit of Cantor we can define a set to be a well-defined collection of objects as long as we furnish a good description/definition of that collection.
As to cardinality: we consider it a property that every set has and that leads to two derived properties of pairs of sets that are defined unambiguously.

Two sets, X and Y, are said to have the same cardinality if “it is possible to put them, by some law, in such a relation to one another that to every element of each one of them corresponds one and only one element of the other” (Cantor, translation by Jourdain). We simply abbreviate this as |X|=|Y|. Additionally we can define what |X|≤|Y| (“the cardinality of X is less than or equal to that of Y”) means that there is a subset Z of Y such that |X|=|Z| (“X has the same cardinality as some subset of Y”).

In this post there are some examples of sets with the same cardinality (provinces of the Netherlands, and months of the year) and with different cardinalities (two teaspoons of chocolate sprinkles).

It is now actually quite straightforward to come to the definition of strictly between. First we define `strictly less’; then `strictly between’ is a combination of twice `strictly less’.

In the example of the chocolate sprinkles the cardinality of the sprinkles in the left hand spoon was strictly less that the cardinality on the right hand side. I paired off the sprinkles on the left with a subset of the sprinkles on the right and it was at once clear that there was no way to pair off both sets with each other. In short, we saw that |L|≤|R| and that |L|≠|R|. And this will be our definition of “the cardinality of X is strictly less than that of Y”, in symbols |X|<|Y|: it is the conjuction of |X|≤|Y| and |X|≠|Y|.

Cantor’s seminal theorem from 1873 can be summarized as |N|<|R|, where N and R denote the sets of natural and real numbers respectively (more on the definition of these in later posts).
Since N is a subset of R it is clear that |N|≤|R|; the hard part of Cantor’s proof was to show that |N|≠|R|, i.e., that there is no way to pair off the natural numbers and the real numbers with each other.

So “the cardinality of Y is strictly between the cardinalities of X and Z” is the conjuction of |X|<|Y| and |Y|<|Z| and ultimately comes down to the following four statements:

  • X can be put into one-to-one correspondentce with a subset of Y,
  • Y can be put into one-to-one correspondentce with a subset of Z,
  • X cannot be put into one-to-one correspondentce with Y, and
  • Y cannot be put into one-to-one correspondentce with Z

For explicitly given sets it is often not too difficult to establish whether this state of affairs holds or not; certainly not with all the tools that Cantor and his followers have developed.

In the case of the Continuum Hypothesis matters lie differently: two of the three sets are there; one should produce the third in the middle, or show that no third exists. At the beginning of the 20th century either possibility probably seemed like an insurmountable task, although Cantor strongly believed in the second alternative.

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