How is cardinality exactly defined as a function and why is it different from the ordinals

Question

Once we construct the definition of the ordinals:

$$0=\{\} \, \, 1=\{0\} \,\, 2=\{0,1\} \,\,3= \{0, 1,2\} \,\, ...$$

And we want to describe the cardinality of the set $S$:

$$S=\{3,2,4\}$$

Intuitively we know that:

$$|S|=3$$

However, how do we describe the function of $\mathbb{card}(x)$? This function must map the Von-Neumann ordinals to the cardinality of the set. How does this function do that? If the cardinality of the set can be described by an ordinal, why do we denote:

$$|\mathbb{N}| = \aleph_0$$

And not:

$$|\mathbb{N}| = \omega_0$$

What would be the ordinal number that would be associated with $\aleph_1$. I thought that it could be $\omega_0 + 1$ since it is the next size of infinity. Like the next size in cardinal arithmetic is always just adding one. Or the idea of the next. However, if it is not that what would it be? Would the construction of the cardinality function help, or not?

If this is not clear, please let me know and I'll edit this a bit more.

Thanks.

Answer 1

As the Wikipedia page notes, there are two ways to approach cardinality. One, which is what you are getting at here, is to construct the cardinal numbers and have a procedure that assigns each set $S$ a unique cardinal $Card(S)$. This construction is somewhat involved and is mostly the domain of set theorists and the like. Most ordinary mathematicians think of cardinality through the relations "$A$ has the same cardinality as $B$" denoted by $|A| = |B|$ and "$A$ has cardinality less than or equal to $B$", denoted by $|A| \leq |B|$.

We define $|A| = |B|$ as $\exists \phi:A \to B, \phi$ is a bijection. And we define $|A| \leq |B|$ by $\exists \phi: A \to B, \phi$ is an injection.

Then the Schroder-Bernstein theorem gives that $|A| \leq |B|$ and $|B| \leq |A| \implies |A| = |B|$.

Now if we just consider finite sets, we can alternatively define a "function" (note it won't be a true set function, since there is no set of all finite sets) $Card(S)$ that assigns a finite set $S$ a unique natural number that is its number of elements. We then can note that $Card(A) = Card(B) \iff |A| = |B|$ and $Card(A) \leq Card(B) \iff |A| \leq |B|$, so these two approaches are the same for finite sets.

Edit: Defining the $Card$ "function" for finite sets. Since $Card$ cannot be a set function as noted above, we are really looking for a predicate $\Phi(A,n)$ s.t. $A$ is finite implies $\exists! n \in \mathbb{N}, \Phi(A,n)$.

Denote $set(n) = \{0,...,n-1\}$. Define $\Phi(A,n) \iff n \in \mathbb{N} \land \exists \phi : A \to set(n), \phi$ is a bijection.

Then to show $\Phi$ has the properties we want, we note that uniqueness comes directly from the nonexistence of bijections between $set(n)$ and $set(m)$ for $n \neq m$ and since $A$ is finite can be defined to mean $\exists n \in \mathbb{N} \exists \phi : A \to set(n), \phi$ is a bijection, we get existence of some $n$ s.t. $\Phi(A,n)$ provided $A$ is finite.

Thus $\Phi$ defines a function, since to each finite $A$, we get a unique $n \in \mathbb{N}$.