It is known that the set of rational numbers are countable.

Union of rational numbers and irrational numbers gives the real numbers.

So \(\displaystyle{\left[{0},{1}\right]}={Q}{\left[{0},{1}\right]}\cup{R}{Q}{\left[{0},{1}\right]}\)

Since Q[0,1] is countable and [0,1] is not countable.

So the only possibility is that \(\displaystyle{R}{Q}{\left[{0},{1}\right]}\) is uncountable.

Hence, the set of irrational numbers in [0,1] is not countable.

Union of rational numbers and irrational numbers gives the real numbers.

So \(\displaystyle{\left[{0},{1}\right]}={Q}{\left[{0},{1}\right]}\cup{R}{Q}{\left[{0},{1}\right]}\)

Since Q[0,1] is countable and [0,1] is not countable.

So the only possibility is that \(\displaystyle{R}{Q}{\left[{0},{1}\right]}\) is uncountable.

Hence, the set of irrational numbers in [0,1] is not countable.