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> You'll never come across a number in reality that you can prove is irrational.

If a square with sides of rational (and non-zero) length can exist in reality, then the length of its diagonal is irrational. So which step along the way isn't possible in reality? Is the rational side length possible? Is the right angle possible?

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They're saying you can't find a ruler accurate enough to be sure the number you measure is sqrt(2) and not sqrt(2) for the first 1000 digits then something else. And eventually, as you build better and better rulers, it will turn out that physical reality doesn't encode enough information to be sure. Anything you can measure is rational.

A perfect mathematical square cannot exist in reality.

Which part is impossible? It was implied that perfect rational numbers are possible, so I’m wondering what stops the square with rational side length from existing and having an irrational diagonal.

An unprovable assertion.

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