Infinity comes in sizes

The whole numbers and real numbers are both infinite, but there is no one-to-one pairing between them.

A set is countably infinite if its members can be paired with the natural numbers. Integers and rational numbers are countable; real numbers are uncountable. Cantor's diagonal argument shows why any proposed list of real numbers in an interval must miss some.

For example: The positive even numbers pair with the positive integers by n → 2n, even though the evens form a proper subset.

Keep in mind: Infinite cardinality behaves differently from finite counting. Claims about the continuum beyond basic uncountability can depend on the axioms being used.

Think about it

Why does the intuition that a proper subset must be smaller work for finite sets but fail in the even-number example?

Connected ideas

Godel and the limits of proof · The problem of induction · Exponential growth and doubling

Sources and further reading

• Stanford Encyclopedia of Philosophy: Set Theory

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