Cantor’s Diagonal Argument
Proof that real numbers are uncountable.
Key idea
Proof that real numbers are uncountable.
In depth
Suppose you could list all real numbers in (0, 1). Build a new number whose nth digit differs from the nth digit of the nth number. It differs from every listed number, so the list is incomplete: the reals are uncountable.
Worked example
If the list starts 0.1234..., 0.5678..., the new number begins 0.2..., 0.6... (changing the first digit of the first number, the second of the second, and so on).
Real-world uses
- Proving that not all infinities are equal
- A classic argument in logic and computer science
Common mistake
Forgetting to avoid digits that cause 0.999... = 1.0 issues by choosing digits carefully.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Cantor’s Diagonal Argument?
Proof that real numbers are uncountable. It is a topic at the expert level in Discrete Math & Number Theory.
Where is Cantor’s Diagonal Argument used in real life?
Real-world examples: Proving that not all infinities are equal; A classic argument in logic and computer science.
What should I learn before Cantor’s Diagonal Argument?
Make sure you are comfortable with Countable and Uncountable Sets, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Cantor’s Diagonal Argument?
Forgetting to avoid digits that cause 0.999... = 1.0 issues by choosing digits carefully.
How can I practise Cantor’s Diagonal Argument?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Cantor’s Diagonal Argument?
Move on to Pigeonhole Principle next.
Related expert topics in Discrete Math & Number Theory
- Strong Induction – Assume all earlier cases to prove the next.
- Well-Ordering Principle – Every nonempty set of positive integers has a least element.
- Partial Orders – Reflexive, antisymmetric, transitive relations.
- Cardinality of Sets – Comparing sizes of infinite sets.
- Countable and Uncountable Sets – ℕ and ℚ are countable; ℝ is not.
- Combinations with Repetition – Stars and bars.