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On the Hamkins approximation property

2006
*
Annals of Pure and Applied Logic
*

According to

doi:10.1016/j.apal.2006.05.005
fatcat:pbcbx4tmlneqdiihqc7mbpdfha
*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] . ...##
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Superstrong and other large cardinals are never Laver indestructible

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 . ...

##
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Fragility and indestructibility II

2015
*
Annals of Pure and Applied Logic
*

' theorems of

doi:10.1016/j.apal.2015.06.002
fatcat:suxxtdrjdvfxjiq3t46oany6fy
*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] . ...##
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Multiversism and Concepts of Set: How Much Relativism Is Acceptable?
[chapter]

2016
*
Boston Studies in the Philosophy of Science
*

Of particular importance will be an account of reference

doi:10.1007/978-3-319-31644-4_11
fatcat:3ptp6udy7racje7z3szftnn7fu
*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. ...##
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The downward directed grounds hypothesis and very large cardinals
[article]

2018
*
arXiv
*
pre-print

Consequently, we establish some fundamental theorems

arXiv:1707.05132v2
fatcat:pbynvlwyfzg2xmlbwel36kvjt4
*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. ...##
###
Certain very large cardinals are not created in small forcing extensions

2007
*
Annals of Pure and Applied Logic
*

See Jech [4],

doi:10.1016/j.apal.2007.07.002
fatcat:coqlsq2xpnd2pdfkn56hjq7mji
*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]. ...##
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The ground axiom is consistent with V $\neq $ HOD

2008
*
Proceedings of the American Mathematical Society
*

Surprisingly, however,

doi:10.1090/s0002-9939-08-09285-x
fatcat:266qcrwjvrefva7zxei57e5ilq
*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. ...##
###
The consistency of level by level equivalence with $V = {\rm HOD}$, the Ground Axiom, and instances of square and diamond

2020
*
Bulletin of the Polish Academy of Sciences Mathematics
*

In

doi:10.4064/ba180529-14-3
fatcat:2rxck3petbhf5nloftwjghioee
*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. ...##
###
Extendible cardinals and the mantle
[article]

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 ...

##
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On ground model definability
[article]

2013
*
arXiv
*
pre-print

These results turn out to have a bearing

arXiv:1311.6789v1
fatcat:daefntlnv5b7pey7ld62iwdive
*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*. ...##
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The Set-theoretic Multiverse : A Natural Context for Set Theory(Mathematical Logic and Its Applications)

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

doi:10.4288/jafpos.19.0_37
fatcat:c7otfm7qxfav3e6zguudzwbwx4
*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*. ...##
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Strongly compact cardinals and the continuum function
[article]

2019
*
arXiv
*
pre-print

We study

arXiv:1901.05313v3
fatcat:5y53nqlet5huxedryzgv4qukfq
*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*. ...##
###
The ground axiom

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*) . ...

##
###
The Ground Axiom (GA)
[article]

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*) . ...

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

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,

doi:10.1007/s00153-009-0146-2
fatcat:o6m4j5t66vbqfl672pwrnlfkze
*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. ...
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