Here's another brain teaser involving infinity...
One afternoon at 4pm, an old man was pulling a bucket of water up from his well, and was delighted to discover that ten golden coins were in the bucket, and they were strangely numbered 1 through 10. He took them inside, and stacked them lovingly in his cellar, and puzzled over their origin.
At 4:30, he returned to the well, and drew the bucket again, and shouted for joy when he saw ten more coins, these labelled 11 through 20. Unbeknownst to him, back in his cellar, a curious mouse had found his stack of coins, and stolen one from the bottom. When the mouse heard the old man returning, he fled to his hole, holding coin number 1. The old man entered the cellar and added coins 11 through 20 to his pile.
At 4:45, he could not resist going out again and checking the well once more, and sure enough, he discovered coins 21 through 30. He hurried inside with them, by which time the little mouse had managed to work coin number 2 out from under the pile. Since there were still 18 coins remaining, the old man didn't notice that any were missing as he put coins 21 to 30 on top of the pile.
The old man made further trips to the well in shorter and shorter times, after 7.5 minutes, then 3.75 minutes, etc, each time making the trip from well to cellar in half the previous time (he was a spry old man), each time adding ten new coins to the top of his ever-growing pile. Also each trip, the mouse managed to creep back out and sneak a single coin from the bottom of the ever growing hoard.
(Take a moment here to convince yourself that infinitely many trips to the well will have occurred by 5pm... and kindly suspend disbelief!)
At 5pm, the old man was very tired, having moved infinitely fast, but was also satisfied that he had gathered the infinitely many numbered coins that the well seemed determined to offer. He returned to his cellar to view his spoils, and was shocked at what he saw!
What did the old man see?
And the really crazy question: What would be different if a sparrow had been stealing coins from the top of the pile instead of a mouse from the bottom. It matters! Think of the numbers on the coins.