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.

Aiming for a contradiction, suppose 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:

 $\ds \exists m \in \N: \,$ $\ds \map \phi {\dfrac q 2}$ $=$ $\ds m$ Definition of Bijection $\ds \leadsto \ \$ $\ds \map \phi q$ $=$ $\ds \map \phi {\dfrac q 2 + \dfrac q 2}$ $\ds$ $=$ $\ds \map \phi {\dfrac q 2} + \map \phi {\dfrac q 2}$ Definition of Semigroup Homomorphism $\ds$ $=$ $\ds m + m$ $\ds$ $=$ $\ds 2 m$ $\ds$ $=$ $\ds n$

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.

$\blacksquare$