scholarly journals Changing the heights of automorphism towers by forcing with Souslin trees over L

2008 ◽  
Vol 73 (2) ◽  
pp. 614-633
Author(s):  
Gunter Fuchs ◽  
Joel David Hamkins
Keyword(s):  
Souslin Trees

AbstractWe prove that there are groups in the constructible universe whose automorphism towers are highly malleable by forcing. This is a consequence of the fact that, under a suitable diamond hypothesis, there are sufficiently many highly rigid non-isomorphic Souslin trees whose isomorphism relation can be precisely controlled by forcing.

Constructible lattices of c-degrees

1982 ◽  
Vol 47 (4) ◽  
pp. 739-754
Author(s):  
C.P. Farrington
Keyword(s):  
Set Theory ◽  
Regular Cardinal ◽  
Boolean Algebras ◽  
Combinatorial Complexity ◽  
Generic Extension ◽  
Transitive Model ◽  
Perfect Set ◽  
Consistency Results ◽  
Combinatorial Principles ◽  
Souslin Trees

This paper is devoted to the proof of the following theorem.Theorem. Let M be a countable standard transitive model of ZF + V = L, and let ℒ Є M be a wellfounded lattice in M, with top and bottom. Let ∣ℒ∣M = λ, and suppose κ ≥ λ is a regular cardinal in M. Then there is a generic extension N of M such that(i) N and M have the same cardinals, and κN ⊂ M;(ii) the c-degrees of sets of ordinals of N form a pattern isomorphic to ℒ;(iii) if A ⊂ On and A Є N, there is B Є P(κ+)N such that L(A) = L(B).The proof proceeds by forcing with Souslin trees, and relies heavily on techniques developed by Jech. In [5] he uses these techniques to construct simple Boolean algebras in L, and in [6] he uses them to construct a model of set theory whose c-degrees have orderlype 1 + ω*.The proof also draws on ideas of Adamovicz. In [1]–[3] she obtains consistency results concerning the possible patterns of c-degrees of sets of ordinals using perfect set forcing and symmetric models. These methods have the advantage of yielding real degrees, but involve greater combinatorial complexity, in particular the use of ‘sequential representations’ of lattices.The advantage of the approach using Souslin trees is twofold: first, we can make use of ready-made combinatorial principles which hold in L, and secondly, the notion of genericity over a Souslin tree is particularly simple.

scholarly journals Closed maximality principles: implications, separations and combinations

2008 ◽  
Vol 73 (1) ◽  
pp. 276-308 ◽  
Author(s):  
Gunter Fuchs
Keyword(s):  
Equivalence Relation ◽  
Regular Cardinal ◽  
Highly Sensitive ◽  
Souslin Trees ◽  
Maximality Principles ◽  
Automorphism Tower

AbstractI investigate versions of the Maximality Principles for the classes of forcings which are <κ-closed, <κ-directed-closed, or of the form Col(κ, <λ). These principles come in many variants, depending on the parameters which are allowed, I shall write MPΓ (A) for the maximality principle for forcings in Γ, with parameters from A. The main results of this paper are:• The principles have many consequences, such as <κ-closed-generic (Hκ) absoluteness, and imply, e.g., that ◊κ holds. I give an application to the automorphism tower problem, showing that there are Souslin trees which are able to realize any equivalence relation, and hence that there are groups whose automorphism tower is highly sensitive to forcing.• The principles can be separated into a hierarchy which is strict, for many κ.• Some of the principles can be combined, in the sense that they can hold at many different κ simultaneously.The possibilities of combining the principles are limited, though: While it is consistent that MP<κ-closed(Hκ +) holds at all regular κ below any fixed α, the “global” maximality principle, stating that MP<κ-closed (Hκ ∪ {κ} ) holds at every regular κ, is inconsistent. In contrast to this, it is equiconsistent with ZFC that the maximality principle for directed-closed forcings without any parameters holds at every regular cardinal. It is also consistent that every local statement with parameters from Hκ⊦ that's provably <κ-closed-forceably necessary is true, for all regular κ.

scholarly journals Higher Souslin trees and the GCH, revisited

2017 ◽  
Vol 311 ◽  
pp. 510-531 ◽  
Author(s):  
Assaf Rinot
Keyword(s):  
Souslin Trees

scholarly journals REDUCED POWERS OF SOUSLIN TREES

2017 ◽  
Vol 5 ◽  
Author(s):  
ARI MEIR BRODSKY ◽  
ASSAF RINOT
Keyword(s):  
Prime Number ◽  
Microscopic Approach ◽  
Tree Construction ◽  
Souslin Trees ◽  
The Relationship ◽  
Single Power

We study the relationship between a $\unicode[STIX]{x1D705}$-Souslin tree $T$ and its reduced powers $T^{\unicode[STIX]{x1D703}}/{\mathcal{U}}$.Previous works addressed this problem from the viewpoint of a single power $\unicode[STIX]{x1D703}$, whereas here, tools are developed for controlling different powers simultaneously. As a sample corollary, we obtain the consistency of an $\aleph _{6}$-Souslin tree $T$ and a sequence of uniform ultrafilters $\langle {\mathcal{U}}_{n}\mid n<6\rangle$ such that $T^{\aleph _{n}}/{\mathcal{U}}_{n}$ is $\aleph _{6}$-Aronszajn if and only if $n<6$ is not a prime number.This paper is the first application of the microscopic approach to Souslin-tree construction, recently introduced by the authors. A major component here is devising a method for constructing trees with a prescribed combination of freeness degree and ascent-path characteristics.

On fake Souslin trees

1969 ◽  
Vol 36 (3) ◽  
pp. 571-573 ◽  
Author(s):  
F. B. Jones
Keyword(s):  
Souslin Trees

scholarly journals Souslin trees and successors of singular cardinals

1986 ◽  
Vol 30 (3) ◽  
pp. 207-217 ◽  
Author(s):  
Shai Ben-David ◽  
Saharon Shelah
Keyword(s):  
Singular Cardinals ◽  
Souslin Trees
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