Ordinals and cardinals

Ordinals and cardinals are ordinary values in omega (yes, infinity is just a fucking value here, have fun), and they are the reason the language has omega and aleph0 as words. This page shows you how to write them, and what arithmetic they support.

Writing them

An ordinal is written omega, epsilon0 or omega1, and ordinals are kept in Cantor normal form. A cardinal is written aleph0, aleph1, aleph or continuum.

Arithmetic on ordinals

+, * and ^ work on ordinals, and the order of the operands matters, as it does in the mathematics (math is a bitch like that). 1 omega + is omega, while omega 1 + is a different ordinal.

1 omega +            w
omega 1 +            w+1
2 omega ^ omega =    1

Subtraction past infinity is an error (you can't subtract your way out of infinity, nice try). card gives the cardinality of an ordinal.

Arithmetic on cardinals

+, * and ^ and the comparisons also work on cardinals.

2 aleph0 ^           2^A0
aleph0 2 aleph0 ^ <  1

aleph1 continuum = is an error and not 0.

Programs

The programs count, ord-demo, powerset, pairing, diagonal and supertask in root/40-infinity.om demonstrate these values, and the tour shows them running.

See also