I am learning classical set theory and how the natural numbers (and other number systems) can be represented by a construction from sets, e.g.:
$$ \underline{0} = \phi $$
$$\underline{1} = \{\phi\}$$
$$\underline{2} = \{\phi, \{\phi\}\} $$
$$\underline{3} = \{ \phi, \{\phi\}, \{\phi, \{\phi\}\} \} $$
$$\vdots$$
And then further work to construct $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$ and $\mathbb{C}$.
Perhaps I got the wrong idea at some point but it seems to be that all sets within ZFC are structures like this over $\phi$.
Equivalently, given set $S$ within ZFC, every chain of containment (where the finiteness of this chain is guaranteed by the Axiom of Regularity i.e. no downward infinite membership chains), teminates with $\phi$:
$$S \ni s_{n-1} \ni s_{n-2} \ni \dots \ni s_{1} \ni \phi $$
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Is that correct?
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If so – is it correct to think of every ZFC set $S$ as equivalent to a directed acyclic graph (DAG), possibly infinite, with source $S$ and sink $\phi$, with a constraint that for every node, each of the children nodes have a unique downstream structure compared to its sibling nodes (to satisfy the Axiom of Extensionality)?
For example, the set $\underline{5}$ is equivalent to the DAG:
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Could we go further and claim: if $f$ is a function from the class of all sets, to the set of all DAGs (with a single sink & single source, such that every node has unique downward structure relative to its siblines, and permitting infinite nodes/edges), as described above, that $f$ is a bijection?
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If I'm correct, then what does this say about more intuitive sets, like the set of all cats in Melbourne, and so on? These sets cannot be expressed within ZFC? (Unless we could find a way to characterize each Melbournian cat and formalize them with structures over $\phi$…)
Best Answer
Technically speaking, Yes. This is because in set theory the structure of the set is usually more important than the elements of the sets themselves because under a function the elements can change greatly, but the set itself is technically the same set. And the empty set is usually the more important set because all other sets can be constructed in such way, this is shown by the universe of al sets $V$, which is constructed as the following.
$$V_0:=\varnothing$$
For ordinals with a predecessor
$$V_{\alpha+1}:=\mathcal{P}(V_{\alpha})$$
For limit ordinals $\lambda$
$$V_\lambda:=\bigcup_{\alpha<\lambda}V_\alpha$$.
And then the complete Universe $V$ Is, $$V:=\bigcup_{\beta}V_\beta$$ This means that the universe of all sets is completely constructed with simply the empty set. Now, this universe $V$ is a fundamental part of the axiom of limitation of size. (It is the axiom that "says" when a class is too big to be a set, which is a whole other thing just by itself.).
However if one wanted to turn these elements ($\varnothing$) to ordinals they could, and this would turn $V$ to the class of all ordinals $Ord$. It all depends on what class you want. [Becuase technically speaking $V$ is the class of all well-founded sets, not of all sets with various elements other than $\varnothing$]
So, technically Yes, But also No.