"Aren't I" is incorrect because you are not supposed to use the word "are" in the first-person singular (present tense) but rather the word "am."
Another example: "I'm still invited to your party, aren't I?" This is WRONG because this person has used "are" in the first-person singular. He or she should have used "am" instead. It would have been correct if this person had instead asked, "I'm still invited to your party, am I not?" or "I'm still invited to your party, right?"
For those who are still unconvinced, remember that I have given grammatical and logical reasons as to why "Aren't I?" is incorrect. If you still want to argue that it is correct just because a lot of people tend to use it then I would point out to you that your argument is an ad populum argument. Just because a lot of people might believe something is correct that does not in and of itself make it correct. A lot of people used to believe the world was flat but that did not make them right. No, they were still wrong, just as people who think "Aren't I?" is correct are still wrong even if there might be a lot of them who think it is correct.