Are some infinities bigger than others?: a maths EPQ idea
A title to start from
Are some infinities bigger than others, and how convincing is Cantor's argument?
Why it works as an EPQ
Countable and uncountable sets can be explained with short, precise proofs, and the resistance Cantor met gives you a debate to weigh.
Scope and difficulty
Solid. Solid. Countability of the rationals, the diagonal argument and the power set theorem are enough; the continuum hypothesis belongs in the conclusion only.
The maths
Builds on these A Level topics: Proof · Algebra and functions.
You would learn:
- Bijections and injections
- Countable and uncountable sets
- The diagonal argument
One possible plan
- Define 'same size' through one-to-one correspondences, with examples such as the even numbers.
- Prove the rationals are countable and the reals are not.
- Present Cantor's power set theorem.
- Evaluate the objections raised at the time and today.
Pitfalls
- Treating infinity as a number in arithmetic.
- A diagonal argument that ignores decimals ending in recurring 9s.
Where to start reading
- Infinity: A Very Short Introduction (Ian Stewart)
- Search for: Cantor diagonal argument decimal expansions 0.999
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