Ten Ways of Looking at Real Numbers

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Every nonstandard real number may be written uniquely as a sum of a standard real number and an infinitesimal.

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#10. Surreal Numbers

Surreal numbers are a subclass of a class of finitely-move two-person games.

One development: a surreal is an ordinal-length sequence of +’s and –’s.

Surreals are lexicographically ordered by -, (empty), +.

The surreal numbers, as a proper class, form an ordered field.

The real numbers are a subfield of the surreals of order .

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#10. Surreal Numbers

Examples of surreal numbers in order:

-- -2

- -1

-+ -1/2

0

+-+ ¾

++++ 4

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#10. Surreal Numbers

Surreals of order :

all dyadic fractions

Surreals of order :

all real numbers

all dyadic fractions

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Many more views of the real numbers

Geometry axioms for the real line

Real numbers as infinite continued fractions

Numeration schemes for real numbers

Alternative foundations: constructivism, intuitionism, nonstandard set theory

Computational approximations to real numbers: floating point numbers, interval arithetic, and so on.

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Many more views of the real numbers

Complexity and randomness measures on real numbers (for example, Turing degrees)

Historical and philosophical perspectives: the real numbers as an idealization of a measurement, the meaning and use of infinitesimals, and so on.

Real numbers as a representation of an infinite sequence of Bernoulli trials

Real numbers generated by formal languages.

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Many more views of the real numbers

Digit patterns in real numbers, such as normal numbers.

Real numbers as set-theoretic codes. A real number may code:

a cardinal collapse

a Borel set

a countable model of set theory

a strategy for an infinite two-person game.

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Organizing Multiple Theories

How should the hypertext on real numbers be organized?

Less than a grand all-encompassing architecture

More that a simple listing of topics in unrelated slots.

The goal is a readable, searchable, general introduction to the real number system.

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Organizing Multiple Theories

It must also show relationships across categories.