Puzzles for Pleasure #2- Hotel Infinity

A long, long time ago in a galaxy far far away, there was a hotel with an infinite number of rooms. The room numbers start at 1 and go on forever...

One day, when every room was occupied, a space pilot on his way to deep space nine (or something) dropped by to spend the night. Even though there was no vacancy, the hotel manager simply notified every guest to move the the room number that was one higher than their current room.

This left room 1 open for the pilot.

The next day, 10 couples on their honeymoon showed up. The hotel manager did the same, only notifying everyone to move to a room number ten higher than their current room, leaving 1 through 10 available to the honeymooners.

The next day, an INFINITE number of guests seek lodging. Is it possible for the hotel manager to accept an infinite number of guests when the hotel already is lodging an infinite number?

TAKE A MOMENT TO THINK ABOUT THAT BEFORE READING ON...THE ANSWER FOLLOWS:

In set theory, no finite set can be put into one to one correspondence with any of its subsets. This means that, if you have a set of 100 colored objects, and within those is a subset of 20 blue objects, you cannot match one blue object to each of the 100 colored objects. Common sense, right?

Well, this is not true for infinite sets. They violate the rule that a whole is greater than any of its individual parts, and an infinite set can be defined as any set which can be put in one to one correspondence with one of it's subsets.

In this case, what the manager should do is move every guest to the room number TWICE as large as their current room number. This moves every one of the infinite guests into an even numbered room, leaving the entire infinite set of odd numbered rooms vacant. Thus, he is able to accept the infinite number of new guests, adding infinity to infinity.

This demonstrates how the infinite set of individuals currently lodging can be placed into one to one correspondence with the subset of all even numbered rooms.

I hope you've enjoyed this installment of puzzles for pleasure. As a bonus, see below:

Can the rational number line (an infinite set consisting of all numbers which are not irrational- numbers like 1.001, 3.11111111111111777773, 4, and 5/3 are rational. Pi and phi are not, because they cannot be expressed as the ratio of two whole numbers) be placed in one to one correspondence with the natural numbers(1, 2, 3, 4, 5)?

Remember, one to one correspondence is matching each member of a set one for one with a member of another set...

To answer this question, you may want to research the concept of Cardinality and Aleph Numbers.

Puzzles for Pleasure #1- The Prediction

I'm going to try something new for a while, I'm going to regularly post paradoxes, puzzles, and logical oddities just to entertain and stimulate! I hope you enjoy...

Can a fortune teller see the future in his crystal ball? 

Let's say a fortune teller, Genie, has a teenage daughter, call her Mary, whom he has promised to buy a car for when she graduates school. 

One day, Mary tells her father that he's a fake who can't tell the future, and challenges him to a test:

She writes something on a piece of paper and locks it into a box. She says, "I've written down an event that will either happen or not happen before sundown." She hands him a blank piece of paper, and says "If the event will happen, write "yes," if it will not, write "no." If you are wrong, buy me a car now, if you are right, don't buy me one at all."

Genie says it's a deal.

He writes something on the card, and at sundown Mary unlocks the box. Genie reads what she wrote:

"Before sundown, you will write NO on the card."

"You've tricked me!" declares Genie. "I wrote YES, so I was wrong, but if I'd written NO I'd be wrong too!"

Mary used her new sports car to drive across country and never went back to her crazy Dad's house.

The original version of this story is about a computer that can only respond "true" or "false" (which is exactly how computers work). We ask it to tell us whether it's next response will be "true," or "false." Since it is not possible for the prediction to be correct, the computer spends ten thousand years thinking and spits out "42," or something like that.

We can reduce this paradox to the question "Will the next word you speak be "no?" Please say "yes" or "no.""

This is a disguised version of the well known "liar" paradox. The liar paradox arises when one states "this sentence is false." It is what is known as a semantic, or truth-value, paradox.

The fun thing about semantic paradoxes is that they can all be rephrased as set theory paradoxes. The sentence, "this statement is false," for example, can be rephrased as "this assertion is a member of the set of all false assertions."

Every semantic paradox has a set theoretical analog and vice versa.

I hope you have enjoyed this first installment of "Puzzles for Pleasure."

Thanks for reading!