Sets, cardinality and the empty set.

Feb 20, 2015

Starting to read Richard Hammack’s Book of Proof, in my life-long quest to understand basic logic and construct valid arguments.

I have read (somewhere) that the empty set, $\emptyset$ is not equal to the set that contains the empty set, $\{ \emptyset \}$. The explanation that Hammack uses (which may not be novel) is:

Consider a set to be the box that contains its elements (maybe I read somewhere a set is only the collection of elements and not some “other” container, anyhow).

• If the box is empty, it is the empty set.
• If the box(2) contains a box(1) then box(2) is not empty, even if box(1) itself is empty.

Another way of thinking about this is say we cannot open the “inner” box(2) and hence cannot tell if box(2) is empty or “full”. Then box(1), when we look inside, certainly contains at least one thing: box(2). Further, box(1) contains only one thing; box(2).

If | A | is the cardinality (size) of set A, then:

|$\emptyset$| = 0

|$\{ \emptyset \}$| = 1

I suppose what we conclude is this: is if sets are the same then they have the same cardinality, therefore if cardinality of two sets is not the same they cannot be the same set.

Another gem I have stumbled across is the Illustrated Book of Bad Arguments, which covers the same ground as David McAndless. I know this is an easy target, but I wish I could deliver an analysis of this quality of logical garbage.

Cardinality and the infinite

This made me realise a “jump” from rational numbers to real numbers. When we are first introduced to sets it is usually by explicitly listing (very small) finite sets such as :{1,2,3,7} or {dog, cat, fish, house!}

We use existing knowledge and methods to say these two sets both have cardinality 4. Further, we have a simple process for telling if an object is the element of a set.

Going to the infinite, countable sets such as the integers { … , -3, -2, -1, 0, 1, 2, 3 …} and the rational numbers using Cantor diagonalisation we can inductively construct any rational number. However, the real numbers can’t be constructed inductively. Does that make them qualitatively different from rational numbers or am I just not smart enough to define a construction method?