Level 6: Countable infinity

Infinity and strange math · level 6 of 8 · rough age 13-14 (a guide for parents; anyone can start anywhere)

Level 5 asked "how close?"; this level asks "how many?". Following Georg Cantor, two sets are the same size when their members can be paired off, and a set is countable when it can be put in one endless list. Step by step you will list the even numbers, the integers, all pairs of numbers, every fraction, and finally every number that solves a polynomial equation, and discover that all of these infinite sets are exactly the same size.

Lesson	You will be able to
Countable sets	Compare sets by pairing, explain what countable means, and list the evens, squares and primes with a position for each
The integers are countable	List all the integers with the zigzag, give its position rule both ways, and zip two countable sets into one list
Pairs are countable	List all pairs of counting numbers along the diagonals, and find the position of any pair and the pair at any position
The rational numbers are countable	List every fraction exactly once, and explain how a set can be dense and countable at the same time
Countable unions of countable sets	Prove that countably many countable sets together are countable, and use it for the hotel, words and programs
Algebraic numbers are countable	Explain what algebraic numbers are, and list them all using a size measure and the fact that polynomials have few roots

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