scholarly journals Subcomplete forcing principles and definable well-orders

2018 ◽  
Vol 64 (6) ◽  
pp. 487-504
Author(s):  
Gunter Fuchs
Keyword(s):  
Subcomplete Forcing

scholarly journals Canonical fragments of the strong reflection principle

2021 ◽  
pp. 2150023
Author(s):  
Gunter Fuchs
Keyword(s):  
Reflection Principle ◽  
Forcing Axioms ◽  
Subcomplete Forcing ◽  
Text Fragment ◽  
Strong Reflection ◽  
Stationary Set ◽  
Mutual Stationarity ◽  
Forcing Axiom ◽  
The Way

For an arbitrary forcing class [Formula: see text], the [Formula: see text]-fragment of Todorčević’s strong reflection principle SRP is isolated in such a way that (1) the forcing axiom for [Formula: see text] implies the [Formula: see text]-fragment of SRP , (2) the stationary set preserving fragment of SRP is the full principle SRP , and (3) the subcomplete fragment of SRP implies the major consequences of the subcomplete forcing axiom. This fragment of SRP is consistent with CH , and even with Jensen’s principle [Formula: see text]. Along the way, some hitherto unknown effects of (the subcomplete fragment of) SRP on mutual stationarity are explored, and some limitations to the extent to which fragments of SRP may capture the effects of their corresponding forcing axioms are established.

scholarly journals SUBCOMPLETE FORCING, TREES, AND GENERIC ABSOLUTENESS

2018 ◽  
Vol 83 (3) ◽  
pp. 1282-1305 ◽  
Author(s):  
GUNTER FUCHS ◽  
KAETHE MINDEN
Keyword(s):  
Forcing Axioms ◽  
First Order ◽  
Generic Absoluteness ◽  
Subcomplete Forcing ◽  
Image Position ◽  
Aronszajn Trees ◽  
Order Structures ◽  
Forcing Axiom

AbstractWe investigate properties of trees of height ω1 and their preservation under subcomplete forcing. We show that subcomplete forcing cannot add a new branch to an ω1-tree. We introduce fragments of subcompleteness which are preserved by subcomplete forcing, and use these in order to show that certain strong forms of rigidity of Suslin trees are preserved by subcomplete forcing. Finally, we explore under what circumstances subcomplete forcing preserves Aronszajn trees of height and width ω1. We show that this is the case if CH fails, and if CH holds, then this is the case iff the bounded subcomplete forcing axiom holds. Finally, we explore the relationships between bounded forcing axioms, preservation of Aronszajn trees of height and width ω1 and generic absoluteness of ${\rm{\Sigma }}_1^1$-statements over first order structures of size ω1, also for other canonical classes of forcing.

HIERARCHIES OF FORCING AXIOMS, THE CONTINUUM HYPOTHESIS AND SQUARE PRINCIPLES

2018 ◽  
Vol 83 (1) ◽  
pp. 256-282 ◽  
Author(s):  
GUNTER FUCHS
Keyword(s):  
Continuum Hypothesis ◽  
Reflection Principle ◽  
Forcing Axioms ◽  
Subcomplete Forcing ◽  
Square Principles ◽  
Consistency Strengths ◽  
Remarkable Cardinals ◽  
Forcing Axiom ◽  
The Continuum

AbstractI analyze the hierarchies of the bounded and the weak bounded forcing axioms, with a focus on their versions for the class of subcomplete forcings, in terms of implications and consistency strengths. For the weak hierarchy, I provide level-by-level equiconsistencies with an appropriate hierarchy of partially remarkable cardinals. I also show that the subcomplete forcing axiom implies Larson’s ordinal reflection principle atω2, and that its effect on the failure of weak squares is very similar to that of Martin’s Maximum.

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