# HIERARCHIES OF FORCING AXIOMS, THE CONTINUUM HYPOTHESIS AND SQUARE PRINCIPLES

@article{Fuchs2018HIERARCHIESOF,
title={HIERARCHIES OF FORCING AXIOMS, THE CONTINUUM HYPOTHESIS AND SQUARE PRINCIPLES},
author={Gunter Fuchs},
journal={The Journal of Symbolic Logic},
year={2018},
volume={83},
pages={256 - 282}
}
• G. Fuchs
• Published 1 March 2018
• Economics
• The Journal of Symbolic Logic
Abstract I 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 at ω 2, and that its effect on the failure of weak squares is…
HIERARCHIES OF (VIRTUAL) RESURRECTION AXIOMS
• G. Fuchs
• Economics
The Journal of Symbolic Logic
• 2018
It is shown that the boldface resurrection axioms for subcomplete or countably closed forcing imply the failure of Todorčević’s square at the appropriate level.
Alternative Cichoń Diagrams and Forcing Axioms Compatible with CH
This dissertation surveys several topics in the general areas of iterated forcing, infinite combinatorics and set theory of the reals. There are two parts. In the first half I consider alternative
SUBCOMPLETE FORCING, TREES, AND GENERIC ABSOLUTENESS
• Environmental Science
The Journal of Symbolic Logic
• 2018
It is shown that subcomplete forcing cannot add a new branch to an ω1-tree, and the relationships between bounded forcing axioms, preservation of Aronszajn trees of height and widthπ1 and generic absoluteness of ${\rm{\Sigma }}_1^1$-statements over first order structures of sizeπ1 are explored.
Forcing axioms and the complexity of non-stationary ideals
• Mathematics
• 2020
We study the influence of strong forcing axioms on the complexity of the non-stationary ideal on $\omega_2$ and its restrictions to certain cofinalities. Our main result shows that the strengthening
Subcomplete forcing principles and definable well‐orders
Enhanced version of bounded forcing axioms are introduced that are strong enough to have the implications of the maximality principles mentioned above.
Canonical fragments of the strong reflection principle
• G. Fuchs
• Environmental Science
J. Math. Log.
• 2021
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:
ARONSZAJN TREE PRESERVATION AND BOUNDED FORCING AXIOMS
• G. Fuchs
• Environmental Science
The Journal of Symbolic Logic
• 2021
A special case of the main result is that for forcing classes that don’t add reals, the three principles at level $2^\omega$ are equivalent.
Iteration theorems for subversions of forcing classes
• Mathematics
• 2020
We prove various iteration theorems for forcing classes related to subproper and subcomplete forcing, introduced by Jensen. In the first part, we use revised countable support iterations, and show
Virtual large cardinals
• Economics
Ann. Pure Appl. Log.
• 2018
COMBINING RESURRECTION AND MAXIMALITY
Abstract It is shown that the resurrection axiom and the maximality principle may be consistently combined for various iterable forcing classes. The extent to which resurrection and maximality

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