If I give you the following definition of the set $A$, how could you prove it is equal the set of the natural numbers without an explicit definiton for the latter?
The set $A$ is inductively defined as follows:
i) $0 \in A$; and
ii) $\forall n$, a natural number, if $n \in A$, then $n+1 \in A$.
I can easily prove that $A$ is contained in the natural numbers, but I'm failing to see how to prove the converse without a similar definition for the natural numbers.
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