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On the Hamkins approximation property

William J. Mitchell
2006 Annals of Pure and Applied Logic  
According to Hamkins [2], a partial ordering P satisfies the δ-approximation property if, whenever A ∈ V P is a subset of an ordinal µ in V P such that In [2, Lemma 13] he proves the following lemma for  ...  The new lemma directly yields Hamkins's newer lemma stating that certain forcing notions have the approximation property.  ...  Then P * Q has the δ-approximation property. This is a generalization of the "Key Lemma" of Hamkins's gap forcing theorems [1, 2] .  ... 
doi:10.1016/j.apal.2006.05.005 fatcat:pbcbx4tmlneqdiihqc7mbpdfha

Superstrong and other large cardinals are never Laver indestructible

Joan Bagaria, Joel David Hamkins, Konstantinos Tsaprounis, Toshimichi Usuba
2015 Archive for Mathematical Logic  
The grounds form a parameterized family Theorem There is a parameterized family { W r | r ∈ V } such that The grounds form a parameterized family Theorem There is a parameterized family { W r | r ∈ V  ...  Theorem (Hamkins) If V ⊆ W has the δ-approximation and δ-cover properties and correct δ + , then V is definable in W .  ...  |B| V < δ. 2 V ⊆ W has δ approximation property if every A ⊆ V with A ∈ W and all small approximations A ∩ a ∈ V , whenever |a| V < δ, is already in the ground model A ∈ V .  ... 
doi:10.1007/s00153-015-0458-3 fatcat:l6mckeph5vbf7brfidsvtthcli

Fragility and indestructibility II

Spencer Unger
2015 Annals of Pure and Applied Logic  
' theorems of Hamkins and Shelah. (3) An answer to a question from our previous paper on the apparent consistency strength of the assertion "The tree property at ℵ 2 is indestructible under ℵ 2 -directed  ...  In this paper we continue work from a previous paper on the fragility and indestructibility of the tree property.  ...  the approximation and covering properties [6] .  ... 
doi:10.1016/j.apal.2015.06.002 fatcat:suxxtdrjdvfxjiq3t46oany6fy

Multiversism and Concepts of Set: How Much Relativism Is Acceptable? [chapter]

Neil Barton
2016 Boston Studies in the Philosophy of Science  
Of particular importance will be an account of reference on the Multiversist conception, and the relativism that it implies.  ...  Multiverse Views in set theory advocate the claim that there are many universes of sets, no-one of which is canonical, and have risen to prominence over the last few years.  ...  move with agility from one model to another." ( [Hamkins, 2012b] , p418) In order to arrive at a first approximation, I shall take this as an initial statement of Hamkins' view.  ... 
doi:10.1007/978-3-319-31644-4_11 fatcat:3ptp6udy7racje7z3szftnn7fu

The downward directed grounds hypothesis and very large cardinals [article]

Toshimichi Usuba
2018 arXiv   pre-print
Consequently, we establish some fundamental theorems on the forcing method and the set-theoretic geology.  ...  For instance, (1) the mantle, the intersection of all grounds, must be a model of ZFC. (2) V has only set many grounds if and only if the mantle is a ground.  ...  Acknowledgements: We would like to thank Joel David Hamkins and Daisuke Ikegami for many fruitful discussions. Some of the ideas that came out of those discussions were used in this paper.  ... 
arXiv:1707.05132v2 fatcat:pbynvlwyfzg2xmlbwel36kvjt4

Certain very large cardinals are not created in small forcing extensions

Richard Laver
2007 Annals of Pure and Applied Logic  
See Jech [4], Hamkins and Woodin [3] and Hamkins [2] for instances of ( * ) for other large cardinal axioms.  ...  Most of the large cardinal axioms from measurable cardinals upwards assert the existence of elementary embeddings j from one transitive set or class to another, where the large cardinal κ is cr( j), the  ...  Lastly, that the δ-approximation property holds is ([2], Lemma 13). For ( * * ), it suffices to show that the δ-approximation property holds for the extension V ⊆ V [G].  ... 
doi:10.1016/j.apal.2007.07.002 fatcat:coqlsq2xpnd2pdfkn56hjq7mji

The ground axiom is consistent with V $\neq $ HOD

Joel David Hamkins, Jonas Reitz, W. Hugh Woodin
2008 Proceedings of the American Mathematical Society  
Surprisingly, however, the Ground Axiom does not hold in all the canonical inner models, for Schindler has observed that the minimal model M 1 of one Woodin cardinal is a forcing extension of one of its  ...  These arguments rely, respectively, on recent work of Laver [5], using methods of Hamkins [3] , and independent work of Woodin [11] , showing that any model of set theory W is first-order definable as  ...  We note that the authors of this article constitute three mathematical generations: Reitz was a dissertation student of Hamkins, who was a dissertation student of Woodin.  ... 
doi:10.1090/s0002-9939-08-09285-x fatcat:266qcrwjvrefva7zxei57e5ilq

The consistency of level by level equivalence with $V = {\rm HOD}$, the Ground Axiom, and instances of square and diamond

Arthur W. Apter
2020 Bulletin of the Polish Academy of Sciences Mathematics  
In the model constructed, there are no restrictions on the class of supercompact cardinals. 2020 Mathematics Subject Classification: 03E35, 03E55.  ...  We construct via forcing a model for the level by level equivalence between strong compactness and supercompactness in which both V = HOD and the Ground Axiom (GA) are true.  ...  The author wishes to thank Gunter Fuchs for helpful conversations on the subject matter of this paper.  ... 
doi:10.4064/ba180529-14-3 fatcat:2rxck3petbhf5nloftwjghioee

Extendible cardinals and the mantle [article]

Toshimichi Usuba
2018 arXiv   pre-print
The mantle is the intersection of all ground models of V. We show that if there exists an extendible cardinal then the mantle is a ground model of V.  ...  κ-covering and the κ-approximation properties.  ...  For every set X, there is r such that W r ⊆ W s for every s ∈ X.A key of the definability of grounds as in Fact 2.1 is the covering and the approximation properties introduced by Hamkins[2]: Definition  ... 
arXiv:1803.03944v3 fatcat:e4lmhfqmonafha7oe7ao2p7via

On ground model definability [article]

Victoria Gitman, Thomas A. Johnstone
2013 arXiv   pre-print
These results turn out to have a bearing on ground model definability for models of ZFC.  ...  It follows from our proof methods that the hereditary size of the parameter that Woodin used to define a ZFC model in its set-forcing extension is best possible.  ...  Laver's proof [Lav07] that ground models of ZFC are definable in their set-forcing extensions uses Hamkins' techniques and results on pairs of models with the δ-cover and δ-approximation properties.  ... 
arXiv:1311.6789v1 fatcat:daefntlnv5b7pey7ld62iwdive

The Set-theoretic Multiverse : A Natural Context for Set Theory(Mathematical Logic and Its Applications)

Joel DAVID HAMKINS
2011 Annals of the Japan Association for Philosophy of Science  
Set・ theorists often take their subject as coiistitut・ing a foundation for thc rcst of mathematics, in the sense that other abstract mathematical objects can  ...  on some of my work in [3] concerning the approximation and cover properties.  ...  considering the fundamental nature of the quest・ion it answers, Laver's proof of this theorem builds on work of mine [3] concerning the approximation and eovering properties.  ... 
doi:10.4288/jafpos.19.0_37 fatcat:c7otfm7qxfav3e6zguudzwbwx4

Strongly compact cardinals and the continuum function [article]

Arthur W. Apter, Stamatis Dimopoulos, Toshimichi Usuba
2019 arXiv   pre-print
We study the general problem of the behaviour of the continuum function in the presence of non-supercompact strongly compact cardinals.  ...  This will follow by a corollary of Hamkins' work of [8] on the approximation and cover properties (which is a generalization of his gap forcing results found in [7] ).  ...  In [6] , Hamkins showed that fast function forcing at an arbitrary strongly compact cardinal adds a function with the Menas property.  ... 
arXiv:1901.05313v3 fatcat:5y53nqlet5huxedryzgv4qukfq

The ground axiom

Jonas Reitz
2007 Journal of Symbolic Logic (JSL)  
The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of  ...  A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set forcing extension of any inner model.  ...  The first are the δ cover and δ approximation properties, formulated by Hamkins [Ham03] , which provide a framework for analyzing extensions and inner models. Definition 5. (Hamkins) .  ... 
doi:10.2178/jsl/1203350787 fatcat:6wfdp4jl4bagtfmblikkalsszy

The Ground Axiom (GA) [article]

Jonas Reitz
2007 arXiv   pre-print
The Ground Axiom is independent of many well-known set-theoretic assertions including the Generalized Continuum Hypothesis, the assertion V=HOD that every set is ordinal definable, and the existence of  ...  A new axiom is proposed, the Ground Axiom, asserting that the universe is not a nontrivial set forcing extension of any inner model.  ...  The first are the δ cover and δ approximation properties, formulated by Hamkins [Ham03] , which provide a framework for analyzing extensions and inner models. Definition 5. (Hamkins) .  ... 
arXiv:math/0609270v2 fatcat:pyy6wvizhngnlcf2upgjte7qvq

Infinite time extensions of Kleene's $${\mathcal{O}}$$

Ansten Mørch Klev
2009 Archive for Mathematical Logic  
Our exposition will presuppose some familiarity with these machines and their theory, comparable to what can be got from reading, for instance, Hamkins and Lewis' papers [2] and [3]; material on Kleene's  ...  Introdution One natural motivation for work on the theory of infinite time Turing machines is the question of how notions and objects from classical computability theory carry over into infinite time.  ...  The author wishes to express his sincerest thanks to Joel David Hamkins for very kind and helpful supervision.  ... 
doi:10.1007/s00153-009-0146-2 fatcat:o6m4j5t66vbqfl672pwrnlfkze
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