Book ChapterDOI
Conjectures of Rado and Chang and Cardinal Arithmetic
Stevo Todorcevic
- pp 385-398
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In this paper, the following conjecture of Richard Rado was studied: a family of intervals of a linearly ordered set is the union of countably many disjoint subfamilies iff every subfamily of size ℵ1 has this property.Abstract:
We study the following conjecture of Richard Rado: a family of intervals of a linearly ordered set is the union of countably many disjoint subfamilies iff every subfamily of size ℵ1 has this property. We connect it with a well-known two-cardinal transfer principle of model theory known as Chang’s Conjecture and show that it solves the Singular Cardinals Problem.read more
Citations
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Journal ArticleDOI
Aronszajn trees and failure of the singular cardinal hypothesis
TL;DR: It is proved from large cardinals that the tree property at λ+ is consistent with failure of the singular cardinal hypothesis at κ, and the two properties are reconciled.
Journal ArticleDOI
Combinatorial dichotomies in set theory
TL;DR: An overview of a research line concentrated on finding toWhich extent compactness fails at the level of first uncountable cardinal and to which extent it could be recovered on some other perhaps not so large cardinal is given.
Journal ArticleDOI
A new Löwenheim-Skolem theorem
Matthew Foreman,Stevo Todorcevic +1 more
TL;DR: In this article, it was shown that any first-order structure has a countable elementary substructure with strong second-order properties, and several consequences for Singular Cardinals Combinatorics are deduced from this.
Journal ArticleDOI
Semimorasses and nonreflection at singular cardinals
TL;DR: Some subfamilies of Pκ(λ), for κ regular, κ ⩽ λ, called (κ, λ)-semimorasses are investigated, and new consistency results involve the existence of nonreflecting objects of singular sizes of uncountable cofinality such as a non reflectinging stationary set in Pκ (λ), a non Reflecting nonmetrizable space of size λ.
Journal ArticleDOI
Open colorings, the continuum and the second uncountable cardinal
TL;DR: In this paper, it was shown that the conjunction of two well-known axioms, OCA [ARS] and OCA[T], implies that the size of the continuum is N 2.
References
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Book ChapterDOI
A decomposition theorem for partially ordered sets
TL;DR: In this article, a partially ordered set P is considered and two elements a and b of P are camparable if either a ≧ b or b ≧ a. If b and a are non-comparable, then they are independent.
Journal ArticleDOI
Forcing axioms and stationary sets
TL;DR: The notion of semi-proper forcing axioms was introduced by Shelah in this article as an obvious strengthening of the Proper Forcing Axiom (PFA), previously formulated and proved consistent by Baumgartner (see [Bal, Del]).
Journal ArticleDOI
Saturation properties of ideals in generic extensions. I
TL;DR: In this paper, the authors consider saturation properties of ideals in models obtained by forcing with countable chain condition partial orderings, and show that for a given cardinal X, if X = (2*°)+ as computed in the ground model, then the answer to the question in paragraph one is "none".