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Axiom 1: All rational numbers can be expressed as a fraction of two integers.

\forall x\in \mathbb{Q} ,\exists \left\{ a,b\right\} \subset \mathbb{Z} :x=\dfrac {a}{b}

Axiom 2 Let l be the property of a fraction being in lowest terms. If both the denominator and numerator are even then the fraction is not in its lowest terms because 2 would be a common factor.

\mathbb{E} \left\{ 2k: k\in\mathbb{Z}\right\}
l\Leftrightarrow a\notin\mathbb{E}\bigvee b\notin\mathbb{E}

Assume it the square root is rational.
\sqrt{2} \in \mathbb{R}
\Rightarrow \sqrt{2}= \frac{a}{b} \because Ax. 1
\Rightarrow 2= \frac{a^2}{b^2}
\Rightarrow 2b^2= a^2
\Rightarrow a^2\in \mathbb{E}

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