Irrational Numbers are Uncountably Infinite

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The set $\R \setminus \Q$ of irrational numbers is uncountable.


From Real Numbers are Uncountable, $\R$ is an uncountable set.

From Rational Numbers are Countably Infinite $\Q$ is countable.

The result follows from Uncountable Set less Countable Set is Uncountable.


Axiom of Choice

This theorem depends on the Axiom of Choice, by way of Uncountable Set less Countable Set is Uncountable.

Because of some of its bewilderingly paradoxical implications, the Axiom of Choice is considered in some mathematical circles to be controversial.

Most mathematicians are convinced of its truth and insist that it should nowadays be generally accepted.

However, others consider its implications so counter-intuitive and nonsensical that they adopt the philosophical position that it cannot be true.