Isomorphism types of Aronszajn trees

Abraham, Uri; Shelah, Saharon

doi:10.1007/bf02761119

Cited by 53 publications

(129 citation statements)

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“…Over time, this conjecture developed in the folklore and at some point it was known to be equivalent -modulo PFA -to the assertion that the above list forms a basis for the class of uncountable linear orders. 1 Moreover this reduction does not require any of the large cardinal strength of PFA. The reader is referred to the final section of [15] for proofs of the above assertions.…”

Section: Introductionmentioning

confidence: 99%

“…A central notion in our analysis is that of the saturation of an Aronszajn tree T -whenever A is a collection of uncountable downward closed subsets of T which have pairwise countable intersection, then A has cardinality at most ω 1 . When possible, 1 The speculation of the consistent existence of a finite basis for the uncountable linear orders seems to have first been made in print in [3], although it still seems to have been unknown at that point that this was equivalent to Shelah's conjecture. 2 The current upper bound is a supercompact cardinal [12].…”

Section: Introductionmentioning

confidence: 99%

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Bounding the consistency strength of a five element linear basis

et al. 2008

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Abstract. 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 can hold in the constructible universe L.A crucial notion in the proof is the saturation of an Aronszajn tree, a statement which may be of broader interest. We show that if all Aronszajn trees are saturated and PFA(ω 1 ) holds, then there is a five element basis for the uncountable linear orders. We show that PFA(ω 2 ) implies that all Aronszajn trees are saturated and that it is consistent to have PFA(ω 1 ) plus every Aronszajn tree is saturated relative to the consistency of a reflecting Mahlo cardinal. Finally we show that a hypothesis weaker than the existence of a Mahlo cardinal is sufficient to force the existence of a five element basis for the uncountable linear orders.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Bounding the consistency strength of a five element linear basis

et al. 2008

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“…Our focus in this paper will be to show that the Proper Forcing Axiom (PFA) implies that any uncountable linear order must contain an isomorphic copy of one of the following five orders: X, ω 1 , ω * 1 , C, and C * . Here X is any fixed set of reals of cardinality ℵ 1 and C is any fixed Countryman line.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1.1 (PFA, [3] PFA is a strengthening of the Baire Category Theorem and is independent of the usual axioms of set theory. Frequently, as in Baumgartner's result above, this axiom can be used to find morphisms between certain structures or to make other combinatorial reductions (see [1], [3], [7], [24], [25], [27]). An additional assumption is necessary in Baumgartner's result because of the following classical construction of Sierpiński.…”

Section: Introductionmentioning

confidence: 99%

A five element basis for the uncountable linear orders

Moore¹

2006

Ann. Math.

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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, ω 1 , ω * 1 , C, C * where X is any suborder of the reals of cardinality ℵ 1 and C is any Countryman line. This confirms a longstanding conjecture of Shelah.

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“…The assumptions of the above theorem are known to be consistent in the case " ν = ω " (see [AS85,Theorem 4.1]). If ν is uncountable, then this assumption implies that (κ) fails and κ is weakly compact in L (see [Tod07,Corollary 6.3.3] and [Tod89, Theorem 3]).…”

Section: ])mentioning

confidence: 99%

Continuous images of closed sets in generalized Baire spaces

Lücke

Schlicht

2015

Isr. J. Math.

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Abstract. Let κ be an uncountable cardinal with κ = κ <κ . Given a cardinal µ, we equip the set κ µ consisting of all functions from κ to µ with the topology whose basic open sets consist of all extensions of partial functions of cardinality less than κ. We prove results that allow us to separate several classes of subsets of κ κ that consist of continuous images of closed subsets of spaces of the form κ µ. Important examples of such results are the following: (i) there is a closed subset of κ κ that is not a continuous image of κ κ; (ii) there is an injective continuous image of κ κ that is not κ-Borel (i.e. that is not contained in the smallest algebra of sets on κ κ that contains all open subsets and is closed under κ-unions); (iii) the statement " every continuous image of κ κ is an injective continuous image of a closed subset of κ κ " is independent of the axioms of ZFC; and (iv) the axioms of ZFC do not prove that the assumption " 2 κ > κ + " implies the statement " every closed subset of κ κ is a continuous image of κ (κ + ) " or its negation.

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Isomorphism types of Aronszajn trees

Cited by 53 publications

References 13 publications

Bounding the consistency strength of a five element linear basis

Bounding the consistency strength of a five element linear basis

A five element basis for the uncountable linear orders

Continuous images of closed sets in generalized Baire spaces

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