Confusion over definition of natural numbers

Question

In my course we have been given the following definition of the natural numbers

The natural numbers $\mathbb{N}$ are the smallest set such that

1. $0\in\mathbb{N}$
2. if $n\in\mathbb{N}$ then $n+1\in\mathbb{N}$

Surely the rational numbers also satisfy this definition. They are the same size as $\mathbb{N}$, have $0\in\mathbb{Q}$ and we can take any $n\in\mathbb{Q}$ and have $n+1\in\mathbb{Q}$.

Is the definition not as rigorous as it could be or are the naturals the smallest set to satisify the definition and I'm using ideas about cardinality incorrectly?

Answer 1

The notion of “smallest set” has nothing to do with cardinality. It refers to set inclusion. No proper subset of the naturals as defined here will satisfy the two conditions.

Answer 2

$ 0\in \Bbb N $ and $ 0 $ is the smallest element. There is no element $ n\in \Bbb N $ such that $ n+1=0 $, and this is not true in $ \Bbb Q$.