η set

In mathematics, an η set is a type of totally ordered set introduced by Hausdorff (1907, p.126, 1914, chapter 6 section 8) that generalizes the order type η of the rational numbers.

Definition

If α is an ordinal then a η_α set is a totally ordered set such that if X and Y are two subsets of cardinality less than ℵ_α such that every element of X is less than every element of Y then there is some element greater than all elements of X and less than all elements of Y.

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 {−1,0,1} 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 a 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 ∑_β<α 2^ℵ_β ≤ ℵ_α.

References

Alling, Norman L. (1962), "On the existence of real-closed fields that are η_α-sets of power ℵ_α.", Trans. Amer. Math. Soc., 103: 341–352, doi:10.1090/S0002-9947-1962-0146089-X, MR 0146089
Chang, Chen Chung; Keisler, H. Jerome (1990) [1973]. Model Theory. Studies in Logic and the Foundations of Mathematics (3rd ed.). Elsevier. ISBN 978-0-444-88054-3.
Felgner, U. (2002), "Die Hausdorffsche Theorie der ηα-Mengen und ihre Wirkungsgeschichte", Hausdorff Gesammelte Werke (PDF), vol. II, Berlin, Heidelberg: Springer-Verlag, pp. 645–674
Hausdorff (1907), "Untersuchungen über Ordnungstypen V", Ber. über die Verhandlungen der Königl. Sächs. Ges. der Wiss. zu Leipzig. Math.-phys. Klasse, 59: 105–159 English translation in Hausdorff (2005)
Hausdorff, F. (1914), Grundzüge der Mengenlehre, Leipzig: Veit & Co
Hausdorff, Felix (2005), Plotkin, J. M. (ed.), Hausdorff on ordered sets, History of Mathematics, vol. 25, Providence, RI: American Mathematical Society, ISBN 0-8218-3788-5, MR 2187098