Level 7: Bigger infinities

Infinity and strange math · level 7 of 8 · rough age 15-16 (a guide for parents; anyone can start anywhere)

In level 6 every infinite set you met could be listed, and so had the same size as the counting numbers. In this level that comfortable picture breaks. Cantor's diagonal argument proves that the real numbers cannot be listed, so infinity comes in more than one size, and Cantor's theorem builds a ladder of bigger and bigger infinities with no top. Along the way you will prove that transcendental numbers exist, explore a dust of points with no length, and see why a line and a square have exactly the same number of points.

Lesson	You will be able to
Cantor's diagonal argument	Carry out the diagonal argument on any list of decimals or 0-1 sequences, and explain why no list of real numbers can be complete
Uncountable means a bigger size	Compare infinite sets with injections and bijections, show that an interval is as big as the whole line, and prove that most real numbers are irrational
Transcendental numbers exist	Show numbers are algebraic, prove that transcendental numbers exist and are uncountable, and explain why proving it for \(\pi\) is much harder
Power sets and Cantor's theorem	Count subsets, prove that every set has more subsets than elements, and explain why there is no largest infinity
The Cantor set	Build the Cantor set, show its length is zero, test points with base 3, and prove it is uncountable
A line and a square have the same number of points	Interleave digits to pack two numbers into one, use the Cantor-Schröder-Bernstein theorem, and explain what dimension really measures

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