Η set explained

In mathematics, an η set (eta set) is a type of totally ordered set introduced by that generalizes the order type η of the rational numbers.

Definition

\alpha

is an ordinal then an

η_\alpha

set is a totally ordered set in which for any two subsets

and

of cardinality less than

\aleph_\alpha

, if every element of

is less than every element of

then there is some element greater than all elements of

and less than all elements of

Examples

The only non-empty countable η₀ set (up to isomorphism) is the ordered set of rational numbers.

Suppose that κ = ℵ_α is a regular cardinal and let X be the set of all functions f from κ to such that if f(α) = 0 then f(β) = 0 for all β > α, ordered lexicographically. Then X is a η_α set. The union of all these sets is the class of surreal numbers.

A dense totally ordered set without endpoints is an η_α set if and only if it is ℵ_α saturated.

Properties

Any η_α set X is universal for totally ordered sets of cardinality at most ℵ_α, meaning that any such set can be embedded into X.

For any given ordinal α, any two η_α sets of cardinality ℵ_α are isomorphic (as ordered sets). An η_α set of cardinality ℵ_α exists if ℵ_α is regular and Σ_β<α 2^ℵ_β ≤ ℵ_α.

References

- Book: Chang . Chen Chung Chang . Chen Chung . Keisler . H. Jerome . Howard Jerome Keisler . Model Theory . 1973 . Elsevier . 3rd . Studies in Logic and the Foundations of Mathematics . 978-0-444-88054-3 . 1990 .
- English translation in