Positive Rational Numbers under Addition not Isomorphic to Natural Numbers

Theorem
The positive rational numbers $\Q_{\ge 0}$ under addition:
 * $\struct {\Q_{\ge 0}, +}$

is not isomorphic to the natural numbers under addition:
 * $\struct {\N, +}$

Proof
From:
 * Positive Rational Numbers under Addition form Commutative Monoid
 * Natural Numbers under Addition form Commutative Monoid

both $\struct {\Q_{\ge 0}, +}$ and $\struct {\N, +}$ form commutative monoids.

there exists an isomorphism $\phi$ from $\struct {\Q_{\ge 0}, +}$ to $\struct {\N, +}$.