Question

Given a list of real numbers, as in the Cantor argument (that is, your list x1, x2, . . . is numbered according to the natural numbers) explain how to construct three different real numbers that you can guarantee are not on the list.

Answer 1

To construct three real numbers not on a given list, alter the nth digit of each nth number on the list in different ways, such as adding or subtracting from it, ensuring that each new number differs from every list number at least at one decimal place.

Step-by-step explanation:

In the context of Cantor's diagonal argument, to construct three different real numbers not on a given list of real numbers, you must ensure that each number you create differs from every number on the list at some decimal place. One common method is to alter the diagonal digits of the real numbers on the list. Here's a step-by-step example:

1. Identify the diagonal: Take the nth digit of the nth number on the list (for each number in the list).
2. Create the first new number: Change each chosen digit to a different digit (other than 9 to avoid issues with repeating 9s).
3. Create the second new number: Alter the digits differently, maybe add 2 to each chosen digit, cycling back to 0 if necessary.
4. Create the third new number: Use another rule of modification, such as subtracting 1 from the chosen digits, with 0 becoming 9 if needed.

Each created number will differ from every number on the list at at least one decimal place, guaranteeing that they are not on the original list.