Uncountable Linear Orders
نویسنده
چکیده
Two important classes of linear orders that are not σ-scattered are the real types and the Aronszajn types: the real types are those uncountable dense linear orders that are separable; the Aronszajn types are those linear orders that are uncountable and yet have no uncountable suborders that are scattered or real types. This latter class was considered—and proved nonempty—long ago by Aronszajn and Kurepa in the context of Souslin’s problem. (The existence of Aronszajn lines was later
منابع مشابه
Uncountable cofinalities of automorphism groups of linear and partial orders
We demonstrate the uncountable cofinality of the automorphism groups of various linear and partial orders. We also relate this to the ‘Bergman’ property, and discuss cases where this may fail even though the cofinality
متن کاملBounding the Consistency Strength of a Five Element Linear Basis
In [13] it was demonstrated that the Proper Forcing Axiom implies that there is a five element basis for the class of uncountable linear orders. The assumptions needed in the proof have consistency strength of at least infinitely many Woodin cardinals. In this paper we reduce the upper bound on the consistency strength of such a basis to something less than a Mahlo cardinal, a hypothesis which ...
متن کاملComputability and uncountable Linear Orders I: Computable Categoricity
We study the computable structure theory of linear orders of size א1 within the framework of admissible computability theory. In particular, we characterize which of these linear orders are computably categorical.
متن کاملA Five Element Basis for the Uncountable Linear Orders
In this paper I will show that it is relatively consistent with the usual axioms of mathematics (ZFC) together with a strong form of the axiom of infinity (the existence of a supercompact cardinal) that the class of uncountable linear orders has a five element basis. In fact such a basis follows from the Proper Forcing Axiom, a strong form of the Baire Category Theorem. The elements are X , ω1,...
متن کاملProper forcing, cardinal arithmetic, and uncountable linear orders
In this paper I will communicate some new consequences of the Proper Forcing Axiom. First, the Bounded Proper Forcing Axiom implies that there is a well ordering of R which is Σ1-definable in (H(ω2),∈). Second, the Proper Forcing Axiom implies that the class of uncountable linear orders has a five element basis. The elements are X, ω1, ω∗ 1 , C, C ∗ where X is any suborder of the reals of size ...
متن کامل