If you restored a ship by replacing each of its wooden parts, would it remain the same ship?
Another classic from ancient Greece, the Ship of Theseus paradox gets at the contradictions of identity. It was famously described by Plutarch:
The ship wherein Theseus and the youth of Athens returned from Crete had thirty oars, and was preserved by the Athenians down even to the time of Demetrius Phalereus, for they took away the old planks as they decayed, putting in new and stronger timber in their places, in so much that this ship became a standing example among the philosophers, for the logical question of things that grow; one side holding that the ship remained the same, and the other contending that it was not the same.
Can an omnipotent being create a rock too heavy for itself to lift?
While we're at it, how can evil exist if God is omnipotent? And how can free will exist if God is omniscient?
These are a few of the many paradoxes that exist when you try to apply logic to definitions of God.
Some people might cite these paradoxes as reasons not to believe in a supreme being; however, others would say they are inconsequential or invalid.
There's an infinitely long "horn" that has a finite volume but an infinite surface area.
Moving ahead to a problem posed in the 17th century, we've got one of many paradoxes related to infinity and geometry.
"Gabriel's Horn" is formed by taking the curve y = 1/x and rotating it around the horizontal axis, as shown in the picture. Using techniques from calculus that make it possible to calculate areas and volumes of shapes constructed this way, it's possible to see that the infinitely long horn actually has a finite volume equal to π, but an infinite surface area.
As stated in the MathWorld article on the horn, this means that the horn could hold a finite volume of paint but would require an infinite amount of paint to cover its entire surface.