Is it possible to explicitly write down a codomain or the range of this 'sequence'?

Question

I came up with questions like the following when reading people discussing the definition of a sequence on freenode IRC ##math channel.

Consider a 'sequence' defined by $f_0=\emptyset$ and $f_1=\{f_0\}=\{\emptyset\}$ and $f_{n+1}=\{f_0,...,f_n\}$ for all n $\in$ $\mathbb{N}$.

Since a sequence is formally defined/considered to be a function, there exist a codomain and the range of $f$.

Question: Is it possible to explicitly write down a set that is a codomain of $f$ or even is the range of $f$ ?

Answer 1

The sequence $\left(f_n\right)_{n\geq 0}$ is precisely the recursive definition of the natural numbers by John von Neumann.

John von Neumann (1923): The embedding of the natural numbers within set theory is \begin{align*} 0:=\emptyset\qquad\qquad n+1:=n\cup\{n\} \end{align*}

$$ $$

We obtain according to this definition \begin{align*} 0= f_0&=\emptyset\\ 1=f_1&=\{0\}\\ &=\{\emptyset\}\\ 2=f_2&=\{0,1\}\\ &=\{\emptyset,\{\emptyset\}\}\\ 3=f_3&=\{0,1,2\}\\ &=\{\emptyset, \{\emptyset\},\{\emptyset,\{\emptyset\}\}\}\\ 4=f_4&=\{0,1,2,3\}\\ &=\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\},\{\emptyset, \{\emptyset\},\{\emptyset,\{\emptyset\}\}\}\}\\ &\cdots \end{align*}

So, we can conclude: The image (range) of $f$ can be identified with the natural numbers $\mathbb{N}$. The codomain of $f$ is any set containing $\mathbb{N}$.