31 found
Order:
  1.  30
    Set-Theoretic Geology.Gunter Fuchs, Joel David Hamkins & Jonas Reitz - 2015 - Annals of Pure and Applied Logic 166 (4):464-501.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   19 citations  
  2.  9
    The Subcompleteness of Magidor Forcing.Gunter Fuchs - 2018 - Archive for Mathematical Logic 57 (3-4):273-284.
    It is shown that the Magidor forcing to collapse the cofinality of a measurable cardinal that carries a length \ sequence of normal ultrafilters, increasing in the Mitchell order, to \, is subcomplete.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  3.  16
    Diagonal Reflections on Squares.Gunter Fuchs - 2019 - Archive for Mathematical Logic 58 (1-2):1-26.
    The effects of the forcing axioms \, \ and \ on the failure of weak threaded square principles of the form \\) are analyzed. To this end, a diagonal reflection principle, \, and it implies the failure of \\) if \. It is also shown that this result is sharp. It is noted that \/\ imply the failure of \\), for every regular \, and that this result is sharp as well.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  4.  7
    Closure Properties of Parametric Subcompleteness.Gunter Fuchs - 2018 - Archive for Mathematical Logic 57 (7-8):829-852.
    For an ordinal \, I introduce a variant of the notion of subcompleteness of a forcing poset, which I call \-subcompleteness, and show that this class of forcings enjoys some closure properties that the original class of subcomplete forcings does not seem to have: factors of \-subcomplete forcings are \-subcomplete, and if \ and \ are forcing-equivalent notions, then \ is \-subcomplete iff \ is. I formulate a Two Step Theorem for \-subcompleteness and prove an RCS iteration theorem for \-subcompleteness (...)
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  5.  32
    A Criterion for Coarse Iterability.Gunter Fuchs, Itay Neeman & Ralf Schindler - 2010 - Archive for Mathematical Logic 49 (4):447-467.
    The main result of this paper is the following theorem: Let M be a premouse with a top extender, F. Suppose that (a) M is linearly coarsely iterable via hitting F and its images, and (b) if M * is a linear iterate of M as in (a), then M * is coarsely iterable with respect to iteration trees which do not use the top extender of M * and its images. Then M is coarsely iterable.
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   8 citations  
  6.  4
    Separating Diagonal Stationary Reflection Principles.Gunter Fuchs & Chris Lambie-Hanson - 2021 - Journal of Symbolic Logic 86 (1):262-292.
    We introduce three families of diagonal reflection principles for matrices of stationary sets of ordinals. We analyze both their relationships among themselves and their relationships with other known principles of simultaneous stationary reflection, the strong reflection principle, and the existence of square sequences.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  7.  8
    Hierarchies of Forcing Axioms, the Continuum Hypothesis and Square Principles.Gunter Fuchs - 2018 - Journal of Symbolic Logic 83 (1):256-282.
    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 atω2, and that its effect on the failure of weak squares is very similar to that of Martin’s Maximum.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  8.  35
    Λ -Structures and s -Structures: Translating the Models.Gunter Fuchs - 2011 - Annals of Pure and Applied Logic 162 (4):257-317.
    I develop a translation procedure between λ-structures, which correspond to premice in the Friedman–Jensen indexing convention on the one hand and s-structures, which are essentially the same as premice in the Mitchell–Steel indexing scheme.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  9.  8
    On Sequences Generic in the Sense of Magidor.Gunter Fuchs - 2014 - Journal of Symbolic Logic 79 (4):1286-1314.
  10.  15
    Closed Maximality Principles: Implications, Separations and Combinations.Gunter Fuchs - 2008 - Journal of Symbolic Logic 73 (1):276-308.
    l 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 $\Sigma _{2}^{1}(H_{\kappa})$ absoluteness, and imply. e.g., that ◇κ holds. I give an application (...)
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark   7 citations  
  11.  41
    Combined Maximality Principles Up to Large Cardinals.Gunter Fuchs - 2009 - Journal of Symbolic Logic 74 (3):1015-1046.
    The motivation for this paper is the following: In [4] I showed that it is inconsistent with ZFC that the Maximality Principle for directed closed forcings holds at unboundedly many regular cardinals κ (even only allowing κ itself as a parameter in the Maximality Principle for < κ -closed forcings each time). So the question is whether it is consistent to have this principle at unboundedly many regular cardinals or at every regular cardinal below some large cardinal κ (instead of (...)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  12.  42
    Degrees of Rigidity for Souslin Trees.Gunter Fuchs & Joel David Hamkins - 2009 - Journal of Symbolic Logic 74 (2):423-454.
    We investigate various strong notions of rigidity for Souslin trees, separating them under ♢ into a hierarchy. Applying our methods to the automorphism tower problem in group theory, we show under ♢ that there is a group whose automorphism tower is highly malleable by forcing.
    Direct download (10 more)  
     
    Export citation  
     
    Bookmark   4 citations  
  13.  9
    Subcomplete Forcing Principles and Definable Well-Orders.Gunter Fuchs - 2018 - Mathematical Logic Quarterly 64 (6):487-504.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  14.  10
    Ehrenfeucht’s Lemma in Set Theory.Gunter Fuchs, Victoria Gitman & Joel David Hamkins - 2018 - Notre Dame Journal of Formal Logic 59 (3):355-370.
    Ehrenfeucht’s lemma asserts that whenever one element of a model of Peano arithmetic is definable from another, they satisfy different types. We consider here the analogue of Ehrenfeucht’s lemma for models of set theory. The original argument applies directly to the ordinal-definable elements of any model of set theory, and, in particular, Ehrenfeucht’s lemma holds fully for models of set theory satisfying V=HOD. We show that the lemma fails in the forcing extension of the universe by adding a Cohen real. (...)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  15.  7
    Subcomplete Forcing, Trees, and Generic Absoluteness.Gunter Fuchs & Kaethe Minden - 2018 - Journal of Symbolic Logic 83 (3):1282-1305.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  16.  7
    Hierarchies of Resurrection Axioms.Gunter Fuchs - 2018 - Journal of Symbolic Logic 83 (1):283-325.
    I analyze the hierarchies of the bounded resurrection axioms and their “virtual” versions, the virtual bounded resurrection axioms, for several classes of forcings. I analyze these axioms in terms of implications and consistency strengths. For the virtual hierarchies, I provide level-by-level equiconsistencies with an appropriate hierarchy of virtual partially super-extendible cardinals. I show that the boldface resurrection axioms for subcomplete or countably closed forcing imply the failure of Todorčević’s square at the appropriate level. I also establish connections between these hierarchies (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  17.  50
    Λ -Structures and s -Structures: Translating the Iteration Strategies.Gunter Fuchs - 2011 - Annals of Pure and Applied Logic 162 (9):710-751.
    Continuing the work of Fuchs [1], I show that the translation functions developed previously map iterable λ-structures to iterable s-structures and vice versa. To this end, I analyse how the translation functions interact with the formation of extender ultrapowers and normal iterations. This analysis makes it possible to translate iterations, and, in a last step, iteration strategies, thus arriving at the result.
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  18.  14
    A Characterization of Generalized Příkrý Sequences.Gunter Fuchs - 2005 - Archive for Mathematical Logic 44 (8):935-971.
    A generalization of Příkrý's forcing is analyzed which adjoins to a model of ZFC a set of order type at most ω below each member of a discrete set of measurable cardinals. A characterization of generalized Příkrý generic sequences reminiscent of Mathias' criterion for Příkrý genericity is provided, together with a maximality theorem which states that a generalized Příkrý sequence almost contains every other one lying in the same extension.This forcing can be used to falsify the covering lemma for a (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   3 citations  
  19.  17
    Successor Levels of the Jensen Hierarchy.Gunter Fuchs - 2009 - Mathematical Logic Quarterly 55 (1):4-20.
    I prove that there is a recursive function T that does the following: Let X be transitive and rudimentarily closed, and let X ′ be the closure of X ∪ {X } under rudimentary functions. Given a Σ0-formula φ and a code c for a rudimentary function f, T is a Σω-formula such that for any equation image ∈ X, X ′ ⊧ φ [f ] iff X ⊧ T [equation image]. I make this precise and show relativized versions of (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  20.  51
    Generic Embeddings Associated to an Indestructibly Weakly Compact Cardinal.Gunter Fuchs - 2010 - Annals of Pure and Applied Logic 162 (1):89-105.
    I use generic embeddings induced by generic normal measures on that can be forced to exist if κ is an indestructibly weakly compact cardinal. These embeddings can be applied in order to obtain the forcing axioms in forcing extensions. This has consequences in : The Singular Cardinal Hypothesis holds above κ, and κ has a useful Jónsson-like property. This in turn implies that the countable tower works much like it does when κ is a Woodin limit of Woodin cardinals. One (...)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  21.  18
    Changing the Heights of Automorphism Towers by Forcing with Souslin Trees Over L.Gunter Fuchs & Joel David Hamkins - 2008 - Journal of Symbolic Logic 73 (2):614 - 633.
    We 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.
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  22.  3
    More on HOD-Supercompactness.Arthur W. Apter, Shoshana Friedman & Gunter Fuchs - 2021 - Annals of Pure and Applied Logic 172 (3):102901.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  23.  48
    Inner-Model Reflection Principles.Neil Barton, Andrés Eduardo Caicedo, Gunter Fuchs, Joel David Hamkins, Jonas Reitz & Ralf Schindler - 2020 - Studia Logica 108 (3):573-595.
    We introduce and consider the inner-model reflection principle, which asserts that whenever a statement \varphi(a) in the first-order language of set theory is true in the set-theoretic universe V, then it is also true in a proper inner model W \subset A. A stronger principle, the ground-model reflection principle, asserts that any such \varphi(a) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width (...)
    Direct download (3 more)  
    Translate
     
     
    Export citation  
     
    Bookmark  
  24.  1
    Aronszajn Tree Preservation and Bounded Forcing Axioms.Gunter Fuchs - 2021 - Journal of Symbolic Logic 86 (1):293-315.
    I investigate the relationships between three hierarchies of reflection principles for a forcing class $\Gamma $ : the hierarchy of bounded forcing axioms, of $\Sigma ^1_1$ -absoluteness, and of Aronszajn tree preservation principles. The latter principle at level $\kappa $ says that whenever T is a tree of height $\omega _1$ and width $\kappa $ that does not have a branch of order type $\omega _1$, and whenever ${\mathord {\mathbb P}}$ is a forcing notion in $\Gamma $, then it is (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  25.  17
    Club Degrees of Rigidity and Almost Kurepa Trees.Gunter Fuchs - 2013 - Archive for Mathematical Logic 52 (1-2):47-66.
    A highly rigid Souslin tree T is constructed such that forcing with T turns T into a Kurepa tree. Club versions of previously known degrees of rigidity are introduced, as follows: for a rigidity property P, a tree T is said to have property P on clubs if for every club set C (containing 0), the restriction of T to levels in C has property P. The relationships between these rigidity properties for Souslin trees are investigated, and some open questions (...)
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  26.  1
    Canonical Fragments of the Strong Reflection Principle.Gunter Fuchs - forthcoming - Journal of Mathematical Logic:2150023.
    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 the forcing axiom for [Formula: see text] implies the [Formula: see text]-fragment of SRP, the stationary set preserving fragment of SRP is the full principle SRP, and 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 (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  27. Canonical Fragments of the Strong Reflection Principle.Gunter Fuchs - forthcoming - Journal of Mathematical Logic.
    For an arbitrary forcing class Γ, the Γ-fragment of Todorčević’s strong reflection principle SRP is isolated in such a way that the forcing axiom for Γ implies the Γ-fragment of SRP, the st...
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  28.  20
    Iteratively Changing the Heights of Automorphism Towers.Gunter Fuchs & Philipp Lücke - 2012 - Notre Dame Journal of Formal Logic 53 (2):155-174.
    We extend the results of Hamkins and Thomas concerning the malleability of automorphism tower heights of groups by forcing. We show that any reasonable sequence of ordinals can be realized as the automorphism tower heights of a certain group in consecutive forcing extensions or ground models, as desired. For example, it is possible to increase the height of the automorphism tower by passing to a forcing extension, then increase it further by passing to a ground model, and then decrease it (...)
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark  
  29.  7
    Incomparable Ω1 -Like Models of Set Theory.Gunter Fuchs, Victoria Gitman & Joel David Hamkins - 2017 - Mathematical Logic Quarterly 63 (1-2):66-76.
    No categories
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  30.  5
    The Solidity and Nonsolidity of Initial Segments of the Core Model.Gunter Fuchs & Ralf Schindler - 2018 - Journal of Symbolic Logic 83 (3):920-938.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark  
  31.  9
    The Stationarity of the Collection of the Locally Regulars.Gunter Fuchs - 2015 - Archive for Mathematical Logic 54 (5-6):725-739.
    I analyze various natural assumptions which imply that the set {ω1L[x]∣x⊆ω}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\{\omega_1^{L[x]} \mid x \subseteq \omega\}}$$\end{document} is stationary in ω1. The focal questions are which implications hold between them, what their consistency strengths are, and which large cardinal assumptions outright imply them.
    No categories
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark