# Classical proof of irrationality of square root of two vs axiomatic construction of the real numbers

It is well known that the greeks proved the irrationality of $\sqrt{2}$ but they do so in a informal manner, i.e. they lack of formal definition of irrational number. Today we have several axiomatic constructions of the reals, and formal definitions of real numbers and irrational numbers, take the Dedekind cuts for example, a irrational number is now defined to be a Dedekind cut that is not a rational cut.

My question is:

"Is the classical proof of the irrationality of $\sqrt{2}$ correct/valid by modern standards?"

If it is not, can we use it's method (with some minor alterations of course) to prove that $\{x \in \mathbb Q : x^2 < 2 \text { or } x < 0 \}$ is not a rational cut ?

• You should define what you mean by the classical proof of the irrationality of $\sqrt 2$, at least the main idea. I would guess that the proof as presented does not stand up, but that the idea can be used to make a good proof. – Ross Millikan Mar 19 '18 at 4:14
• Yes it's valid. The ancients showed that if $m,n\in \Bbb N$ then $m^2\ne 2n^2,$ and therefore there cannot exist $x\in \Bbb Q$ such that $x^2=2.$ – DanielWainfleet Mar 19 '18 at 11:22