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 , 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.

(please give mathematical explaination too not just english)

problem 1 : Cantor's proof assumes you can create some table of all the numbers in [0,1) and use it to reach a contradiction. However, that's just some table, not all tables, and the fact that some table can't enumerate [0,1) doesn't mean that all tables can't, because ()[() ] doesn't necessarily imply ()[() ] in predicate logic. So, Cantor's proof is invalid because it violates the rules of predicate logic.

problem 2: In Cantor's proof, the diagonal digits in the assumed table of all numbers in [0,1) are used to produce a magic number that should be in the table yet is somehow missing. However, you can apply the same construction to an assumed table of all numbers in instead: just produce a magic number which is a natural number that differs on its first digit from the first natural number in the table, on its second digit

from the second number, on its third from the third, and so on. This would imply that the natural numbers are uncountable as well, but that's nonsense since they are the counting numbers by definition. Therefore, the diagonalization technique itself must be fundamentally

invalid, and so is Cantor's proof.