Formal Definition of Cardinality

Question

I am a first year student. I am trying to find a precise definition of cardinal numbers. I am unable to find it anywhere. I am fine with finite cardinalities. I also know that different sets have different sizes. But how to have a precise definition of a cardinal number ?

Answer 1

Well, since you have just started exploring mathematics, here is a (pretty simple) definition due to Frege.

You consider the "collection" of all sets. Put an equivalence relation on this as follows :

Sets $A$ and $B$ are equivalent iff there is a bijection $A \xrightarrow{\sim} B$.

It is an exercise to check that this is indeed an equivalence relation.

Now, a cardinal is defined to be an equivalence class under this relation. So for instance, the equivalence class containing $\{1,2,...,n\}$ is the cardinal $n$. The class containing $\mathbb{N}$ is the cardinal Aleph-Naught. The class containing $\mathbb{R}$ is the cardinal $c$ (continuum) and so on.

Answer 2

While Category_Theorist's answer is a good idea to have in mind about how you could define cardinal numbers, in the standard foundations for mathematics (ZFC) there are technicalities that prevent it from working. The problem is that the equivalence class of all sets of a given cardinality is not actually a set--it is a proper class, a collection which is too big to actually be a set.

There are various ways to circumvent this difficulty, all of which are unfortunately pretty technical if you are just starting to learn set theory. The standard approach is to, instead of taking the equivalence class of all sets of a given cardinality, take one particular representative of the equivalence class. In the context of ZFC, it turns out that there is a natural and convenient canonical representative set of each cardinality. The idea is to restrict to sets which are well-ordered by the element relation $\in$. These sets are called ordinals, and by the rigidity properties of well-orderings, the collection of ordinals themselves is well-ordered by $\in$, with each ordinal being the set of all smaller ordinals. You can then define a cardinal number to be an ordinal which is the least ordinal of its cardinality. So, for example, the cardinal number $\aleph_0$ is the least infinite ordinal (which is the set of all finite ordinals), and the cardinal number $\aleph_1$ is the least uncountable ordinal (which is the set of all countable ordinals).

This approach only captures all possible cardinalities if you know that every set can be well-ordered, so that it can be put into bijection with some ordinal. This statement is equivalent to the axiom of choice, so if you do not assume the axiom of choice, you need a different definition for cardinals. Here the standard approach is to fall back on the idea of taking the equivalence class of sets of a given cardinality, but instead of taking all sets of a given cardinality (which would form a proper class instead of a set), you take only the sets of minimal possible rank of that cardinality, which do form a set. (The rank of a set is an ordinal number that roughly captures how many steps it takes to build that set starting from the empty set.)

Answer 3

Eric Wofsey has already given a nice answer for the precise definition of cardinal numbers, but let me point out that for most math one actually does not need these "canonical representatives", instead one can work with the cardinality relation, so even if we do not have a notion of cardinal number, we can say that #$\mathbb{N}$<#$\mathbb{R}$, since there is an injection from $\mathbb{N}$ to $\mathbb{R}$, but no bijection.

I point this out because the formal definition of cardinal number requires some advanced set theory.