ordinal arithmetic

Another example are the ordinals under the usual operations of ordinal arithmetic.

The cardinal and ordinal arithmetic are reviewed.

Due to the focus on countable ordinals, ordinal arithmetic is used throughout, except where otherwise noted.

In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation.

This established the richness of the hierarchy of infinite sets, and of the cardinal arithmetic and ordinal arithmetic that Cantor had defined.

We can apply this, for example, to the class of limit ordinals: the -th ordinal, which is either a limit or zero is (see ordinal arithmetic for the definition of multiplication of ordinals).

In the first section, Conway provides an axiomatic construction of numbers and ordinal arithmetic, namely, the integers, reals, the countable infinity, and entire towers of infinite ordinals, using a notation that is essentially an almost trite (but critically important) variation of the Dedekind cut.

He applied the same idea to prove Cantor's theorem: the cardinality of the power set of a set A is strictly larger than the cardinality of A. This established the richness of the hierarchy of infinite sets, and of the cardinal and ordinal arithmetic that Cantor had defined.