η set

Short description: Type of totally ordered set

In mathematics, an η set (eta 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 [math]\displaystyle{ \alpha }[/math] is an ordinal then an [math]\displaystyle{ \eta_\alpha }[/math] set is a totally ordered set in which for any two subsets [math]\displaystyle{ X }[/math] and [math]\displaystyle{ Y }[/math] of cardinality less than [math]\displaystyle{ \aleph_\alpha }[/math], if every element of [math]\displaystyle{ X }[/math] is less than every element of [math]\displaystyle{ Y }[/math] then there is some element greater than all elements of [math]\displaystyle{ X }[/math] and less than all elements of [math]\displaystyle{ Y }[/math].

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 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

• 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 
• Chang, Chen Chung; Keisler, H. Jerome (1990). 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, II, Berlin, Heidelberg: Springer-Verlag, pp. 645–674, http://hausdorff-edition.de/media/pdf/Eta_Alpha.pdf 
• 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, https://archive.org/details/grundzgedermen00hausuoft 
• Hausdorff, Felix (2005), Plotkin, J. M., ed., Hausdorff on ordered sets, History of Mathematics, 25, Providence, RI: American Mathematical Society, ISBN 0-8218-3788-5, https://books.google.com/books?id=M_skkA3r-QAC