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.
Thanks for taking the time to read me.
Best Answer
It seems to me that your 'proof' that the set $A \subseteq \mathbb{N}$ must have a mistake because basically you're just defining what Tom Apostol calls an inductive set in his Calculus book. I mean, $A$ can just be a set of real numbers and could perfectly well satisfy your two propierties.
For instance you can take $A = [0, \infty[$ and then $A$ satisfies both properties, but nonetheless $A$ is not contained in $\mathbb{N}$.
In fact Apostol's book defines $\mathbb{N}$ as the set of all numbers that belong to every inductive set.