In Cantor's diagonalization argument, why is it guaranteed that the constructed number d is not equal to rₙ for any n?
ABecause d is irrational and all listed numbers are rational
BBecause d was constructed to differ from rₙ specifically in its nth decimal digit
CBecause d has a different number of digits than any rₙ
DBecause d lies outside the interval [0,1]
The construction is the key: d's nth digit is chosen to differ from the nth digit of rₙ specifically. So d ≠ r₁ (differs in digit 1), d ≠ r₂ (differs in digit 2), and in general d ≠ rₙ (differs in digit n). The diagonal structure ensures that every element of the supposed complete list is ruled out at a different, designated position — this is what makes the argument work.
Question 2 Multiple Choice
A skeptic says: 'The diagonalization argument only shows one number is missing. Just add d to the list and now it's complete.' What is the correct response?
AThe argument fails if you extend the list — diagonalization only works for finite lists
BThe new list also contains a missing element, constructible by the same diagonalization procedure applied to the new list
Cd cannot be added to the list because it is not a real number
DThe argument actually shows infinitely many numbers are missing, so adding one doesn't help
The argument is a proof by contradiction about any supposed complete list — not just one particular list. If you add d to get a new list r₁, r₂, …, rₙ, d, rₙ₊₁, …, the same diagonalization procedure applied to this new list produces yet another number d' not on the new list. No matter how you extend or rearrange the list, the construction always finds a missing element. This is why it proves uncountability, not just the absence of a specific number.
Question 3 True / False
The diagonalization argument works by finding a specific real number that was accidentally omitted from the list.
TTrue
FFalse
Answer: False
The argument constructs a number guaranteed to be missing — it does not find a pre-existing omission. Starting from any supposed complete list, it systematically builds d to differ from every listed element at a specific position. The number d depends on the list itself; it is deliberately engineered to escape the list, not discovered independently. This constructive aspect is what makes the proof so powerful and general.
Question 4 True / False
Cantor's diagonalization argument generalizes beyond real numbers: for any set A, the power set P(A) has strictly larger cardinality than A.
TTrue
FFalse
Answer: True
The generalization is exact. Given any function f: A → P(A), define a set D = {a ∈ A : a ∉ f(a)} — this is the diagonal construction in set-theoretic form. D differs from f(a) for every a ∈ A (since a ∈ D iff a ∉ f(a)), so D is not in the range of f, meaning f cannot be surjective. No bijection between A and P(A) can exist. Applying this to ℕ, P(ℕ), P(P(ℕ)), … produces an infinite hierarchy of distinct infinities.
Question 5 Short Answer
Why does the diagonalization argument specifically use the diagonal positions (a₁₁, a₂₂, a₃₃, …) rather than some other selection of positions from the list?
Think about your answer, then reveal below.
Model answer: The diagonal positions are the unique choice that allows one construction to simultaneously target every element on the list. By using position n to target rₙ, the construction ensures d ≠ rₙ for every n using a single, unified rule. Any other selection would fail to cover all elements: if you used position 1 to differ from every rₙ, you could only guarantee d ≠ r₁. The diagonal is the clever design that makes the argument simultaneous rather than sequential.
This is why 'diagonalization' is such a distinctive technique — it is not just proof by contradiction, but a specific geometric insight about how to construct a single object that escapes an entire infinite list at once. Understanding this shows why the same idea appears in Gödel's incompleteness proof and the halting problem: the diagonal construction is a general tool for building self-referential objects that escape any enumeration.