Death by infinity puzzles and the Axiom of Choice

TL;DR

Using the concept of infinity, this video explores the perfect murder, a puzzle of guessing hat colors, and the implications of the Axiom of Choice in mathematics.

Transcript

Welcome to the first Mathologer video of the year. Today is all about playing with infinity and the infamous Axiom of Choice to commit the perfect murder and to cheat death. Because I need the whole width of the screen for most of the video you won't see much of me except now and at the end of the video. Bit of an experiment. Okay, so let's get sta... Read More

Key Insights

• 💯 Infinity can be used conceptually to create scenarios with logical impossibilities, such as the perfect murder where none of the assassins are responsible for the victim's death.
• 😚 Memorized sequences and the concept of being "close" based on differences in binary digits can be used in the puzzle scenario to increase the chances of the assassins walking free.
• 😫 The Axiom of Choice, while controversial, plays a role in mathematics by allowing for the selection of elements from collections of sets and enabling the exploration of paradoxical theorems.

Questions & Answers

Q: How does the first scenario demonstrate the concept of the perfect murder?

In the first scenario, the perfect murder occurs because none of the assassins actually killed Batman, despite his certain death. By creating an infinite loop of assassins, it becomes logically impossible for any one assassin to be the killer.

Q: How does the second scenario involve a puzzle and the Axiom of Choice?

In the second scenario, the assassins have to guess their own hat color based on the hats worn by others. By using memorized sequences and the concept of being "close" in terms of the differences in binary digits, they can come up with a strategy that ensures only finitely many of them get killed. The Axiom of Choice is implied in the selection of one sequence from each box.

Q: How does the third scenario change the strategy and increase their chances of survival?

In the third scenario, the assassins are arranged in a line facing different directions. They have to guess their own hat color one by one. By counting the differences in the hat colors they see, they can determine their own hat color and ensure that everyone except the first assassin walks free.

Q: Why is the Axiom of Choice controversial in mathematics?

The Axiom of Choice allows for the selection of one element from each set in a collection of non-empty sets. This axiom is controversial because it leads to paradoxical and counterintuitive results, such as the Banach-Tarski paradox, where a solid ball can be split into parts and reassembled to create two solid balls of the same size.

Summary & Key Takeaways

• In the first scenario, an evil mastermind uses infinitely many assassins to ensure the death of Batman, but none of them are actually responsible for his death.

• In the second scenario, the assassins have a chance to walk free if they can correctly guess the color of their own hat based on the hats worn by others, using a strategy based on memorized sequences.

• In the third scenario, the assassins can use a different strategy and ensure that everyone except the first assassin walks free by counting the differences of hat colors they see.