Question

Identify fallacy in Cantors proof

Construct ℝ[0,1)as follows. For each natural number, produce a real number by concatenating adigit0, a decimal point, and the reverse of the digits in that natural number. Enumerate the resulting real numbers in a tablelike so:

This shows thatℝ[0,1)can be enumerated.Cantor’s proof depends on this interval being uncountable, so theproof is invalid.We could enumerate the other intervals similarly to show that all of ℝis countable.

Answer 1

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Solution ::

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The fallacy lies in the fact that all natural numbers have finite representation, while all real numbers do not.

Consider the real number \small \pi , which is an irrational number. Thus, it's decimal representation is non terminating and non repeating. Now, we have the first few digits of

\large {\pi\over 4}=0.78539816397......

which lies in the interval [0,1)

However, in our enumeration, we take each natural number and reverse it's digits and put them after a 0 and a decimal point to get a corresponding real number. So, the first number after the decimal is the last digit of the natural number, and the last number is the first digit of the natural number. But, irrational number do not have finite decimal representations, moreover, the last digit of \small \pi , or even, \small \pi /4 is not even known. So, All natural numbers have finite representation, and the digits after the decimal of \small \pi /4 do not terminate, nor do they repeat. So, the corresponding natural number, according to the enumeration used, cannot have a finite representation either, which contradicts the fact that it is a natural number. Hence, the given proof has a fallacy.