cardinality and equivalent classes

Question

The algebra book I'm reading defines cardinality as:

The cardinality of a set $A$ is the equipotency class to which $A$ belongs to. It is indicated with $Card(A)$.

Two sets A and B are called equipotent if there exists a 1-1 mapping of A onto B. The equipotency class is composed by all the sets satisfying this condition.

The definition usually found of cardinality for example https://en.wikipedia.org/wiki/Cardinality

is that of the "number of elements of the set". I'm confused by the two notions I don't understand whether the cardinality is a set as defined in the book or a number. In case the two definitions are equivalent, if you can show me a way to prove it.

Answer 1

The equivalence relation for cardinality is bijection. That is, $A \sim B$ if there is a bijection between $A$ and $B$.

You have described an equivalence relation among elements of $A$. But cardinality is an equivalence relation among sets (in the universe of sets).

Answer 2

The two uses of the word are very closely related.

If you prefer to think of cardinality as a "number" then choose from your equivalence class a canonical representative. In the case of finite sets, that canonical representative might be of the form $\{0,1,2,3,4,\dots,n-1\}$ for an $n$ element set, which has the added benefit of being exactly how the number $n$ is defined if we were going with the Von Neumann method of defining the natural numbers.

Now, when we start talking about infinite sets, we can't so easily point to a "number" that describes the size of the set so easily and so it is common to hear "The cardinality of the natural numbers" or "The cardinality of the continuum." You might consider the use of ordinals to circumvent that, but here too it is common to talk about a representative of the equivalence class. Note that talking about a representative from the equivalence class is just as good as talking about the equivalence class as a whole. They both get across the same information when speaking.

So, the end result of the answer of "is cardinality a 'number' or a 'set'" you might say that it is both.

Answer 3

We are not talking about an equivalence relation on $A$. We are talking about an equivalence relation on $V$, the class of all sets. If $A$ and $B$ are any two sets, then we say that $A\sim B$ if there exists a bijection (a one-to-one correspondence) from $A$ to $B$. It’s easy to show that the relation $\sim$ is an equivalence relation on $V$. And then we define the cardinality of a set $A$ to be the class of all sets $X$ such that $A\sim X$

This is a good definition, because it is easy to show that the cardinality of $A$ equals the cardinality of $B$ if and only if $A\sim B$. This exactly corresponds to the intuitive idea of “the number of elements in a set”. If you have a bunch of coconuts and a bunch of bananas, then the number of coconuts is equal to the number of bananas if and only if you can pair off the coconuts and bananas in a one to one correspondence.

And now we can define $0$ to be the cardinality of the empty set, i.e. the class of all sets that can be put in bijection with the empty set, i.e. the set containing the empty set. And we can define $1$ to be the cardinality of the set containing the empty set, i.e. class of all sets that can be put in bijection with the set containing the empty set i.e. the class of all singleton sets. And so on.

Note that this way of defining cardinal numbers is kind of old-fashioned; it was used by people like Gottlob Frege in the 1800’s, but it’s not used as much anymore. What is used more nowadays is the von Neuman definition, where we define $0$ to be the empty set $1$ to be the set containing the empty set, etc. But in my opinion the Frege definition makes the notion of cardinality clearer than the von Neumann definition. More for details see Gottlob Frege’s book “Foundations of Arothmetic.”