Infinity

What Infinity means

In mathematics infinity is not one giant number but a family of sizes, compared by pairing rather than counting: two collections are the same size if every member of one can be matched with exactly one member of the other. The whole numbers and the even numbers pair off perfectly, so they are the same size. The points on a line cannot be paired with the whole numbers, so there are more of them: a proof, not a feeling. Owning this changes what we let "infinite" imply. We stop assuming that endless means complete, that more of something forever must contain every possibility. We start asking the pairing question: can the possibilities be listed, and does the endless thing cover the list? Before, infinite meant everything. After, we know there are endless collections that miss almost everything.

Infinity examples: where it shows up

• At work: every computer stores numbers with a fixed number of digits, so nearly every real quantity it holds is a rounded stand-in; the odd penny that will not reconcile lives in that gap.
• With money: interest compounded "continuously" is a limit of ever smaller steps, and the limit is a number with no last digit, which is why the rate on the statement is always an approximation.
• In the news: "in an infinite universe everything happens" and "given infinite time the monkeys write Shakespeare" both assume endless means all; the pairing test shows whether that follows, and often it does not.

Where it sits

Open to walk. See where Infinity sits on the sky →

Nearest on the sky

• Combinatorial Explosion
• Fractals
• Gödel's Incompleteness
• Recursion
• The Halting Problem
• Symmetry

A filled dot marks a star that is already open to walk.