TITLE: Direct proof of irrationality?
QUESTION [35 upvotes]: There are plenty of simple proofs out there that $\sqrt{2}$ is irrational. But does there exist a proof which is not a proof by contradiction? I.e. which is not of the form: 
Suppose $a/b=\sqrt{2}$ for integers $a,b$. 
[deduce a contradiction here] 
$\rightarrow\leftarrow$, QED
Is it impossible (or at least difficult) to find a direct proof because ir-rational is a negative definition, so "not-ness" is inherent to the question? I have a hard time even thinking how to begin a direct proof, or what it would look like. How about: 

$\forall a,b\in\cal I \;\exists\; \epsilon$ such that $\mid a^2/b^2 - 2\mid > \epsilon$.

REPLY [3 votes]: Most common axiom systems I've seen are a list of $\forall$ and $\exists$ axioms. If you look at a minimal underlying logic, most of the common rules for transforming these axioms shouldn't change the $\forall$ or $\exists$ into a $\neg \exists$. So you could fix a logic system, and argue that the only method that results in a $\neg \exists$ statement is the equivalent of proof by contradiction.
You haven't fixed a logic system in your original question, but the "proof by direct substitution" method won't be sufficient.