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

By definition of isomorphism:
 * $\phi$ is a homomorphism
 * $\phi$ is a bijection.

Let $n \in \N$ be odd.

Let $q \in \Q_{\ge 0}$ such that $\map \phi q = n$.

Such a $q$ exists and is unique by definition of bijection.

But then we have:

But this contradicts the assertion that $n$ is odd.

So by Proof by Contradiction there can be no such isomorphism from $\struct {\Q_{\ge 0}, +}$ to $\struct {\N, +}$.

Hence the result.