Madame Maths Paradoxical Puzzlers

Welcome, puzzle enthusiast! We've got a brain-twisting, reality-bending collection of paradoxical puzzles for the most adventurous of maths whizzes.

The Barber Paradox

Imagine a barber who's also a mathematician, and he's trying to solve this paradoxical puzzle:

This one's a doozy:

A man walks into a library and asks the librarian, "Do you have any books on the Liar Paradox?"
The librarian replies, "It's a matter of fact, and this is a library."
Is the man a liar or the librarian?
What does it even mean, anyway?

Time travel, you say? Well, imagine you're a time traveler who meets your grandfather as a young man.

You tell him, "Hey, grandpa, you're gonna die in a tragic accident."
But if you tell him, and he dies, then you're never born, and you never get to travel back in time to tell him...
Is the paradox resolved or just delayed?