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- Natural numbers are built from zero and a successor function, with addition and multiplication defined recursively.
- Integers are encoded as ordered pairs of naturals representing positive and negative components, with equivalence classes based on addition.
- Rational numbers are pairs of integers representing fractions, with equivalence based on multiplication; the set of rationals is countable.
- Computable numbers are approximations by algorithms (Turing machines), and they are also countable.
- Real numbers are defined via Dedekind cuts on rationals, forming an uncountable continuum that includes uncomputable numbers.
- Most real numbers are uncomputable, and the cardinality of reals is strictly greater than that of naturals, proven by Cantor's diagonal argument.