History of mathematics · Maths EE idea · Solid
Cantor and the sizes of infinity
A research question to start from
How did Cantor show that the rationals are countable but the reals are not, and why were his arguments controversial?
A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.
Why it works as a maths EE
Two short, deep proofs worked through, with the debate as context for discussion.
Mathematics you would need
- Bijections
- Countability
- The diagonal argument
- Decimal representations
Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.
One possible line of attack
- Prove the rationals are countable.
- Prove the reals are uncountable, handling decimals that end in repeating 9s.
- Discuss the reaction and what it shows about proof.
Scope and difficulty
Solid. Solid.
Pitfalls
- Philosophy without proofs.
- Ignoring the 0.999… issue.
Where to start reading
Search a library catalogue or a university's open lecture notes for: Cantor diagonal argument proof; countability of rationals. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).