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 ηα set is a totally ordered set such that if X and Y are two subsets of cardinality less than ℵα 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=ℵα 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 ηα set. The union of all these sets is the class of surreal numbers.

A dense totally ordered set without endpoints is a ηα set if and only if it is ℵα saturated.

Properties

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

For any given ordinal α, any two ηα sets of cardinality ℵα are isomorphic (as ordered sets). An ηα set of cardinality ℵα exists if ℵα is regular and ∑β<α 2ℵβ ≤ ℵα.

References
  • Alling, Norman L. (1962), "On the existence of real-closed fields that are ηα-sets of power ℵα.", 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" (PDF), Hausdorff Gesammelte Werke, 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, 25, Providence, RI: American Mathematical Society, ISBN 0-8218-3788-5, MR 2187098
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.