I have a question regarding Cantor's proof that the set of irrational numbers is uncountable. As far as I know, to prove this, Cantor proves that there exists a mapping between irrationals and naturals so that at least one irrational is left out. However, the existence of such a mapping does not imply the nonexistence of a one-to-one mapping.
Did I understand the proof wrong? If yes, an explanation would be much appreciated!
Thank you for taking the time to read my question!
You are mistaken. Cantor proved that given any function $f$ from the natural numbers to the irrationals, there is an irrational number which is not in the range of $f$. This number will be different given different functions, but this shows that there cannot be any surjecitve function from $\Bbb N$ onto $\Bbb{R\setminus Q}$.