Infinite sets
When a set never runs out, what can “how many” still mean, and are all infinite sets the same size?
Where you are 4 inside
What this area is about
A finite set is counted by matching it with \{1, 2, \ldots, n\}. An infinite set has no last number to count to, but matching still works: two sets have the same size when there is a bijection between them. Taking that as the definition gives surprises. An infinite set can be the same size as part of itself, and not every infinite set is the same size as \mathbb{N} = \{0, 1, 2, \ldots\}.
How these fit together
The definition. Matching two sets one to one defines “same size” by a bijection.
The first size of infinity. Countable sets are those that are finite or can be put in one endless list. The integers and the rationals are countable.
Results that go beyond it. The diagonal argument proves that \mathbb{R} is uncountable: any list of real numbers misses one. Cantor’s theorem proves that no set A is the same size as its power set \mathcal{P}(A), so there is no largest infinity.
A path through it
The lessons form a chain, each using the one before: matching sets one to one, then countable sets, then the diagonal argument, then Cantor’s theorem. Before starting, the area needs bijections and the cardinality of a finite set; the diagonal argument also uses proof by contradiction, and Cantor’s theorem uses power sets.