Question

Cantor's proof is often referred to as "Cantor's diagonalization argument." Explain why this is a reasonable name.

Answer 1

Cantor's proof, also known as Cantor's diagonalization argument, is a proof in set theory that shows the existence of uncountably infinite sets. The name 'Cantor's diagonalization argument' is appropriate because the proof involves constructing a diagonal sequence by choosing elements from a given list in a specific way.

Step-by-step explanation:

Cantor's proof, also known as Cantor's diagonalization argument, is a proof in set theory that shows the existence of uncountably infinite sets. The name 'Cantor's diagonalization argument' is appropriate because the proof involves constructing a diagonal sequence by choosing elements from a given list in a specific way.

Here is a step-by-step explanation of Cantor's diagonalization argument:

1. Assume that there exists a list that contains all the real numbers between 0 and 1.
2. Construct a new number by taking the first digit of the first number in the list, the second digit of the second number, the third digit of the third number, and so on.
3. Change each digit to a different digit (for example, if the digit is 1, change it to 2; if the digit is 2, change it to 3, and so on).
4. The resulting number cannot be in the original list because it differs from each number in the list in at least one digit.
5. Therefore, the assumption that such a list exists leads to a contradiction. Hence, there is no list that can contain all the real numbers between 0 and 1, proving the existence of uncountably infinite sets.