12  Paradoxes and Contradictions: Puzzles that present seemingly contradictory or paradoxical situations.

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12.1 Bertrand’s paradox

📖 A paradox that arises from the set theory, which states that the set of all sets that do not contain themselves cannot contain itself.

12.1.1 Problem

In the case of Bertrand’s paradox, if we assume that the set of all sets that do not contain themselves does contain itself, what happens to the paradox?

• Hint:
  • Consider the logical implications of both possibilities.
• Answer:
  • If the set of all sets that do not contain themselves contains itself, then it must also not contain itself, leading to a contradiction.

12.1.2 Problem

Can a set of all sets that contain themselves be constructed without violating Bertrand’s paradox?

• Hint:
  • Think about the nature of self-reference and the limits of set theory.
• Answer:
  • No, such a set cannot be constructed without violating Bertrand’s paradox, as it would lead to a logical contradiction.

12.1.3 Problem

If Bertrand’s paradox is true, does that mean that set theory is incomplete or inconsistent?

• Hint:
  • Consider the implications of the paradox for the foundations of mathematics.
• Answer:
  • Bertrand’s paradox highlights the limitations of set theory and raises questions about its completeness and consistency, but it does not necessarily imply that it is incomplete or inconsistent.

12.1.4 Problem

Can Bertrand’s paradox be resolved by introducing a new type of set that does not follow the usual rules of set theory?

• Hint:
  • Explore the possibility of extending the concept of a set.
• Answer:
  • It is possible to introduce new types of sets, such as paraconsistent sets or fuzzy sets, that may be able to resolve Bertrand’s paradox, but this requires extending the foundations of set theory.

12.1.5 Problem

What are the broader philosophical implications of Bertrand’s paradox?

• Hint:
  • Consider the implications for the nature of truth, logic, and the limits of human knowledge.
• Answer:
  • Bertrand’s paradox raises profound philosophical questions about the nature of truth, logic, and the limits of human knowledge, highlighting the complexities of self-reference and the paradoxes that can arise in formal systems.

12.2 Liar’s paradox

📖 A paradox that arises from the statement ‘this statement is false’.

12.2.1 Problem

If someone says ‘I am lying’, and they are telling the truth, what are they?

• Hint:
  • Consider the implications of the statement on itself.
• Answer:
  • A liar

12.2.2 Problem

If someone says ‘I am lying’, and they are lying, what are they?

• Hint:
  • Think about the logical consequences of the statement.
• Answer:
  • Also a liar

12.2.3 Problem

If someone says ‘The next statement is false’, and the next statement is ‘The previous statement is true’, what can you conclude?

• Hint:
  • Analyze the relationship between the two statements.
• Answer:
  • Both statements cannot be true simultaneously, leading to a paradox.

12.2.4 Problem

If a barber shaves everyone in town who does not shave themselves, who shaves the barber?

• Hint:
  • Consider the barber’s actions in relation to themselves.
• Answer:
  • The barber shaves himself if he does not shave himself, and he does not shave himself if he shaves himself, leading to a paradox.

12.2.5 Problem

If a statement is both true and false, what is it?

• Hint:
  • Explore the nature of truth and falsehood.
• Answer:
  • A paradox

12.3 Sorites paradox

📖 A paradox that arises from the gradual change in properties of an object, at which point it ceases to be considered the same object.

12.3.1 Problem

If a single grain of sand does not make a pile, and adding one grain at a time will never create a pile, at what point does a collection of grains become a pile?

• Hint:
  • Consider the gradual accumulation of grains.
• Answer:
  • There is no definitive point at which a collection of grains becomes a pile.

12.3.2 Problem

If a ship is built with wooden parts, and over time each wooden part is replaced with a metal part, at what point does the ship cease to be the same ship?

• Hint:
  • Consider the gradual replacement of parts.
• Answer:
  • The ship remains the same ship even if all of its parts have been replaced, as long as it retains its original identity.

12.3.3 Problem

If you have a glass of water and you add a drop of water to it every minute, at what point does the glass become full?

• Hint:
  • Consider the gradual increase in the amount of water.
• Answer:
  • The glass never becomes full, as the amount of water added each minute is infinitesimal.

12.3.4 Problem

If you have a pile of sand and you remove one grain of sand at a time, at what point does the pile cease to be a pile?

• Hint:
  • Consider the gradual decrease in the number of grains.
• Answer:
  • The pile remains a pile even if one grain is removed, as long as it retains enough grains to be considered a pile.

12.3.5 Problem

If you have a piece of paper and you fold it in half repeatedly, at what point does it become too thick to fold?

• Hint:
  • Consider the gradual increase in the thickness of the paper.
• Answer:
  • The paper becomes too thick to fold after a finite number of folds, as the thickness of the paper doubles with each fold.

12.4 Grandfather paradox

📖 A paradox that arises from the possibility of time travel, where a person travels back in time and kills their own grandfather.

12.4.1 Problem

If you could go back in time to kill your grandfather before your father was born, would you still exist?

• Hint:
  • Consider the consequences of your actions.
• Answer:
  • If you kill your grandfather, your father would never be born, and therefore you would not exist.

12.4.2 Problem

If you travel back in time and give your younger self a winning lottery ticket, who really won the lottery?

• Hint:
  • Think about the origin of the lottery ticket.
• Answer:
  • You won the lottery, because you were the one who bought the ticket and gave it to your younger self.

12.4.3 Problem

If you go back in time and prevent your parents from ever meeting, what happens?

• Hint:
  • Consider the paradox of your own existence.
• Answer:
  • You would never have been born, so the paradox is impossible.

12.4.4 Problem

If you could go back in time and kill Hitler as a baby, would you?

• Hint:
  • Think about the potential consequences of changing history.
• Answer:
  • This is a moral dilemma with no easy answer. Killing Hitler could prevent millions of deaths, but it could also create a new timeline with unknown consequences.

12.4.5 Problem

If you travel back in time and give yourself a message, what would you say?

• Hint:
  • Consider what advice you would find most valuable.
• Answer:
  • This is a personal question, but some possible messages include words of encouragement, warnings about future events, or instructions on how to live a better life.

12.5 Ship of Theseus paradox

📖 A paradox that arises from the gradual replacement of parts of an object, at which point it is considered a different object.

12.5.1 Problem

A man makes a promise to his friend that he will give him his grandfather’s axe. Many years later, the man gives his friend an axe. He tells his friend that this is his grandfather’s axe, but the handle and the head have both been replaced multiple times over the years.

• Hint:
  • What makes something the same object over time, even if the parts are replaced?
• Answer:
  • The axe is still considered the same object, even though the parts have been replaced, because it retains its identity and function.

12.5.2 Problem

A woman owns a beautiful pearl necklace that has been passed down through her family for generations. One day, the clasp breaks and she takes it to a jeweler to be fixed. The jeweler replaces the clasp, but in the process, he accidentally drops the necklace and the pearls scatter all over the floor. The jeweler carefully collects all of the pearls and restrings them, but when he gives the necklace back to the woman, she is furious. Why?

• Hint:
  • What makes an object unique and irreplaceable?
• Answer:
  • The woman is upset because, even though the necklace has been repaired, it is no longer the same necklace. The original necklace had sentimental value because it had been passed down through her family. The new necklace, even though it is made of the same pearls, does not have the same history or meaning.

12.5.3 Problem

A man builds a boat out of wood. Over time, the wood rots and he replaces the planks one by one. Eventually, every single plank has been replaced. Is it still the same boat?

• Hint:
  • What is the essence of an object?
• Answer:
  • This is known as the Ship of Theseus paradox. There is no definitive answer, but it raises questions about the nature of identity and change.

12.5.4 Problem

A woman buys a new car. Over the next few years, she replaces the engine, the transmission, the brakes, the tires, and the body panels. Is it still the same car?

• Hint:
  • What makes something the same thing over time?
• Answer:
  • Like the Ship of Theseus paradox, there is no easy answer to this question. Some people might say that the car is still the same because it retains its identity and function. Others might say that it is a different car because all of the major components have been replaced.

12.5.5 Problem

A man has a grandfather clock that has been in his family for generations. One day, the clock stops working and he takes it to a clockmaker to be repaired. The clockmaker replaces the movement, the pendulum, and the case. When the man gets the clock back, he is disappointed to find that it is no longer the same clock. Why?

• Hint:
  • What makes an object unique and meaningful?
• Answer:
  • The man is disappointed because, even though the clock is now working again, it is no longer the same clock that his grandfather owned. The original clock had sentimental value because it had been in his family for generations. The new clock, even though it is identical in appearance and function, does not have the same history or meaning.