Question : for each proof, identify at least one fallacy and explain in detail why the fallacy leads to a contradiction. For these problems, Cantor's proof refers to the use of the table diagonalization technique to show that the real numbers are uncountable (as presented in Example 23 of Section 4.1 in the textbook), countability and (d)enumerability are equivalent, the notation [0,1) represents the set of real numbers in the interval from 0 to 1 exclusive of 1, and the magic number is a number produced by the diagonalization technique that contradicts the completeness of any assumed table of real numbers.

Problem 1: Cantor's proof shows that we can always produce some magic number as a real number that is missing from the assumed table of all numbers in [0,1) , contradicting the completeness of the table. However, just add to the table. Now that is present in the table, there is no longer any contradiction, and the table contains every real number, rendering Cantor's proof invalid.

Problem 2: The numbers in [0,1) can be simply enumerated starting with 0, then the smallest number larger than 0, then the smallest number larger than that number, then the smallest number larger than that number, and so on, ending with the largest number strictly less than 1. Since this is an enumeration of [0,1) , the set is countable, so Cantor's proof is wrong to conclude that it is uncountable.