Souslin Hypothesis Does Not Imply “Every Aronszajn Tree is Special”

1982 ◽  
pp. 278-303
Author(s):  
Saharon Shelah
Keyword(s):  
Aronszajn Tree

scholarly journals SQUARES, ASCENT PATHS, AND CHAIN CONDITIONS

2018 ◽  
Vol 83 (04) ◽  
pp. 1512-1538 ◽  
Author(s):  
CHRIS LAMBIE-HANSON ◽  
PHILIPP LÜCKE
Keyword(s):  
Partial Order ◽  
Set Theory ◽  
Consistency Strength ◽  
Chain Conditions ◽  
Regular Cardinals ◽  
Aronszajn Tree ◽  
Square Principles ◽  
Image Position ◽  
Aronszajn Trees ◽  
Consistency Strengths

AbstractWith the help of various square principles, we obtain results concerning the consistency strength of several statements about trees containing ascent paths, special trees, and strong chain conditions. Building on a result that shows that Todorčević’s principle $\square \left( {\kappa ,\lambda } \right)$ implies an indexed version of $\square \left( {\kappa ,\lambda } \right)$, we show that for all infinite, regular cardinals $\lambda < \kappa$, the principle $\square \left( \kappa \right)$ implies the existence of a κ-Aronszajn tree containing a λ-ascent path. We then provide a complete picture of the consistency strengths of statements relating the interactions of trees with ascent paths and special trees. As a part of this analysis, we construct a model of set theory in which ${\aleph _2}$-Aronszajn trees exist and all such trees contain ${\aleph _0}$-ascent paths. Finally, we use our techniques to show that the assumption that the κ-Knaster property is countably productive and the assumption that every κ-Knaster partial order is κ-stationarily layered both imply the failure of $\square \left( \kappa \right)$.

scholarly journals A forcing axiom for a non-special Aronszajn tree

2020 ◽  
Vol 171 (8) ◽  
pp. 102820
Author(s):  
John Krueger
Keyword(s):  
Aronszajn Tree ◽  
Forcing Axiom

scholarly journals The special Aronszajn tree property

2019 ◽  
Vol 20 (01) ◽  
pp. 2050003 ◽  
Author(s):  
Mohammad Golshani ◽  
Yair Hayut
Keyword(s):  
Regular Cardinal ◽  
Generic Extension ◽  
Proper Class ◽  
Tree Property ◽  
Supercompact Cardinals ◽  
Aronszajn Tree ◽  
Aronszajn Trees

Assuming the existence of a proper class of supercompact cardinals, we force a generic extension in which, for every regular cardinal [Formula: see text], there are [Formula: see text]-Aronszajn trees, and all such trees are special.

Large discrete parts of the E-tree

1988 ◽  
Vol 53 (3) ◽  
pp. 980-984 ◽  
Author(s):  
Harold Simmons
Keyword(s):  
Standard Model ◽  
Peano Arithmetic ◽  
Unique Root ◽  
First Order ◽  
Aronszajn Tree ◽  
The Standard Model

Let PA be first order Peano arithmetic, let Λ be the lattice of Π1 sentences modulo PA, and let S be the poset of prime filters of Λ ordered by reverse inclusion. We show there are large convex discrete parts of S; in particular there are convex parts which form a completed Baire tree or an Aronszajn tree.The elements of S, which we call nodes, correspond to the extensions of PA which are complete for sentences. Equivalently, for each model of PA the Π1-theory ∀() of is a node, and every node occurs in this form. Note that the Π1-theory ∀() of the standard model (i.e. the filter of true Π1 sentences) is the unique root of S.This poset S, which is sometimes called the E-tree, was first studied in [1] where it is shown that:(1) The poset is tree-like, i.e. the set of predecessors of any node is linearly ordered.(2) The poset has branches, each of which is closed under unions and intersections; in particular each branch has a maximum member.(3) There are branches on which ∀() does not have an immediate successor. Further properties of the E-tree are given in [2]−[7]. In particular in [4] Misercque shows that:(4) There are branches on which ∀() does have an immediate successor.(5) There are nodes with both an immediate predecessor and an immediate successor.The two results (3) and (4) show that there are fundamentally different branches of S, and (5) shows that parts of branches may be discrete.

SQUARE WITH BUILT-IN DIAMOND-PLUS

2017 ◽  
Vol 82 (3) ◽  
pp. 809-833 ◽  
Author(s):  
ASSAF RINOT ◽  
RALF SCHINDLER
Keyword(s):  
Regular Cardinal ◽  
Infinite Cardinal ◽  
List Type ◽  
Type Number ◽  
Aronszajn Tree ◽  
Combinatorial Principles ◽  
Souslin Trees ◽  
Diamond Principle

AbstractWe formulate combinatorial principles that combine the square principle with various strong forms of the diamond principle, and prove that the strongest amongst them holds inLfor every infinite cardinal.As an application, we prove that the following two hold inL:1.For every infinite regular cardinalλ, there exists a special λ+-Aronszajn tree whose projection is almost Souslin;2.For every infinite cardinalλ, there exists arespectingλ+-Kurepa tree; Roughly speaking, this means that this λ+-Kurepa tree looks very much like the λ+-Souslin trees that Jensen constructed inL.

Creatures on ω1 and weak diamonds

2009 ◽  
Vol 74 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Heike Mildenberger
Keyword(s):  
Aronszajn Tree ◽  
Aronszajn Trees

AbstractWe specialise Aronszajn trees by an ωω-bounding forcing that adds reals. We work with creature forcings on uncountable spaces.As an application of these notions of forcing, we answer a question of Moore, Hrušák and Džamonja whether implies the existence of a Souslin tree in a negative way by showing that “ and every Aronszajn tree is special” is consistent relative to ZFC.

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