7.4. Implications of Cantor's Theorem

Try these first

1. 1.

   Define cardinality.

   Hint

   Think about how we count elements.

2. 2.

   What is a power set?

   Hint

   Consider subsets including the empty set.

3. 3.

   What does Cantor's theorem state?

   Hint

   Think about how a set relates to its subsets.

4. 4.

   True or False: The power set of any set always has the same cardinality as the original set.

   • True
   • False

   Hint

   Consider what the theorem states regarding cardinality.

5. 5.

   Consider a different infinite set B with a countable cardinality. Construct a proof using Cantor's diagonalization principle to show there are more numbers between 0 and 1 than the set of natural numbers.

   Hint

   Focus on how to create a number that deviates from each decimal place.

6. 6.

   Discuss the philosophical implications of Cantor's theorem for concepts of infinity in mathematics and science.

   Hint

   Think about how different infinities can apply in various real-world scenarios or mathematical theories.