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Strongly adequate sets and adding a club with finite conditions

2013
*
Archive for Mathematical Logic
*

We present

doi:10.1007/s00153-013-0358-3
fatcat:qilunpmbwnakzosvjwmlfwkbca
*a*forcing poset for*adding**a**club*to*a*fat stationary subset of ω 2*with**finite**conditions*, thereby showing that*a*version of the forcing posets of Friedman [1]*and*Mitchell [3] for*adding**a*... We continue the study of*adequate**sets*which we began in [2] by introducing the idea of*a**strongly**adequate**set*, which has an additional requirement on the overlap of two models past their comparison point ... See Neeman [4] for*a*simplification of some of the arguments for*adding**a**club*to ω 2*with**finite**conditions*. ...##
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Adding a club with finite conditions, Part II

2014
*
Archive for Mathematical Logic
*

We define

doi:10.1007/s00153-014-0406-7
fatcat:bq3ecooe45etjpyjmnnlml2kl4
*a*forcing poset which adds*a**club*subset of*a*given fat stationary*set*S ⊆ ω 2*with**finite**conditions*, using S-*adequate**sets*of models as side*conditions*. ... This construction, together*with*the general amalgamation results concerning S-*adequate**sets*on which it is based, is substantially shorter*and*simpler than our original version in [3] . ... The theory of*adequate**sets*introduced in [2] provides*a*framework for*adding*generic objects on ω 2*with**finite**conditions*using countable models as side*conditions*. ...##
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Coherent adequate forcing and preserving CH

2015
*
Journal of Mathematical Logic
*

We show that there exists

doi:10.1142/s0219061315500051
fatcat:m3lvu6bfbferxiokyanazdt5ki
*a*forcing poset for*adding**a**club*subset of ω 2*with**finite**conditions*while preserving CH, solving*a*problem of Friedman [3]. ... We develop*a*general framework for forcing*with*coherent*adequate**sets*on H(λ) as side*conditions*, where λ ≥ ω 2 is*a*cardinal of uncountable cofinality. ...*Adding**a**club*preserving CH*A*forcing poset for*adding**a**club*to*a*fat subset of ω 2 using*adequate**sets*as side*conditions*appears in [6] . ...##
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Coherent Adequate Forcing and Preserving CH
[article]

2014
*
arXiv
*
pre-print

We show that there exists

arXiv:1406.3302v1
fatcat:tjmvubowwvh6rcf7ctxlhl42fe
*a*forcing poset for*adding**a**club*subset of ω_2*with**finite**conditions*while preserving CH, solving*a*problem of Friedman. ... We develop*a*general framework for forcing*with*coherent*adequate**sets*on H(λ) as side*conditions*, where λ>ω_2 is*a*cardinal of uncountable cofinality. ...*Adding**a**club*preserving CH*A*forcing poset for*adding**a**club*to*a*fat subset of ω 2 using*adequate**sets*as side*conditions*appears in [4] . ...##
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QUOTIENTS OF STRONGLY PROPER FORCINGS AND GUESSING MODELS

2016
*
Journal of Symbolic Logic (JSL)
*

Specifically, we prove that quotients of forcing posets which have universal

doi:10.1017/jsl.2015.46
fatcat:jmb7i4u5rnfp3luuavikvadg2q
*strongly*generic*conditions*on*a*stationary*set*of models by certain nice regular suborders satisfy the ω 1-approximation property ... We prove that*a*wide class of*strongly*proper forcing posets have quotients*with*strong properties. ... Recall the forcing poset P S for*adding**a**club*subset of S*with**finite**conditions*([1, Theorem 3]). ...##
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Quotients of Strongly Proper Forcings and Guessing Models
[article]

2015
*
arXiv
*
pre-print

Specifically, we prove that quotients of forcing posets which have simple universal

arXiv:1406.3306v2
fatcat:jky47xnjpje27bb2pam7c6sbji
*strongly*generic*conditions*on*a*stationary*set*of models by certain nice regular suborders satisfy the ω_1-approximation ... We prove that*a*wide class of*strongly*proper forcing posets have quotients*with*strong properties. ... Recall the forcing poset P S for*adding**a**club*subset of S*with**finite**conditions*([1, Theorem 3]). ...##
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Coherent Adequate Sets and Forcing Square
[article]

2014
*
arXiv
*
pre-print

We introduce the idea of

arXiv:1406.3309v1
fatcat:oh6q7ibyh5hxxalr5qbjcg53u4
*a*coherent*adequate**set*of models, which can be used as side*conditions*in forcing. ... As an application we define*a*forcing poset which adds*a*square sequence on ω_2 using*finite**conditions*. ... Forcing Square*with**Finite**Conditions*We define*a*forcing poset which adds*a*square sequence*with**finite**conditions*, using coherent*adequate**sets*as side*conditions*. ...##
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Coherent adequate sets and forcing square

2014
*
Fundamenta Mathematicae
*

We introduce the idea of

doi:10.4064/fm224-3-5
fatcat:dn5k5bc6vjce3bnjlhdpt2yhke
*a*coherent*adequate**set*of models, which can be used as side*conditions*in forcing. ... As an application we define*a*forcing poset which adds*a*square sequence on ω 2 using*finite**conditions*. ... In previous work [3] we introduced the idea of an*adequate**set*of models*and*showed how to use*adequate**sets*as side*conditions*in forcing*with**finite**conditions*. ...##
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Forcing with Adequate Sets of Models as Side Conditions
[article]

2016
*
arXiv
*
pre-print

We present

arXiv:1406.3307v3
fatcat:e7n2s3vv6nftdlmx6a6szwl23e
*a*general framework for forcing on ω_2*with**finite**conditions*using countable models as side*conditions*. ... We give several examples of this type of forcing, including*adding**a*function on ω_2,*adding**a*nonreflecting stationary subset of ω_2 ∩cof(ω),*and**adding*an ω_1-Kurepa tree. ... Friedman ([3] )*and*Mitchell ([10] , [11] ) independently lifted this method up to ω 2 by showing how to add*a**club*subset of ω 2*with**finite**conditions*. ...##
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Mitchell's Theorem Revisited
[article]

2016
*
arXiv
*
pre-print

In this paper we give

arXiv:1506.01983v3
fatcat:v4aawtn5oza3topqpbs3zj4quq
*a*new proof of Mitchell's theorem, deriving it from an abstract framework of side*condition*methods. ... Mitchell's theorem on the approachability ideal states that it is consistent relative to*a*greatly Mahlo cardinal that there is no stationary subset of ω_2 ∩cof(ω_1) in the approachability ideal I[ω_2] ...*A*basic tool in the proof is*a*forcing poset for*adding**a**club*subset of ω 2*with**finite**conditions*, using*finite**sets*of countable models as side*conditions*. ...##
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Proper forcing remastered
[article]

2013
*
arXiv
*
pre-print

We then discuss some applications: the Friedman-Mitchell poset for

arXiv:1110.0610v3
fatcat:agafxndjjbhxvnlw7h64urpg2i
*adding**a**club*in \omega_2*with**finite**conditions*, Koszmider's forcing construction of*a*strong chain of length \omega_2 of functions from ... In these notes we present the method introduced by Neeman of generalized side*conditions**with*two types of models. ...*Adding**a**club*in ω 2*with**finite**conditions*We now present*a*version of the Friedman-Mitchell (see [3]*and*[7] ) forcing for*adding**a**club*to ω 2*with**finite**conditions*. ...##
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The Approachability Ideal Without a Maximal Set
[article]

2018
*
arXiv
*
pre-print

We develop

arXiv:1607.04772v3
fatcat:nmekousx3bdi3nxe2lw6yqblpa
*a*forcing poset*with**finite**conditions*which adds*a*partial square sequence on*a*given stationary*set*,*with**adequate**sets*of models as side*conditions*. ... We then develop*a*kind of side*condition*product forcing for simultaneously*adding*partial square sequences on multiple stationary*sets*. ... Later, Krueger [6] developed*a*forcing poset for*adding**a*square sequence*with**finite**conditions*using the framework of coherent*adequate**sets*. ...##
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Exactly controlling the non-supercompact strongly compact cardinals
[article]

2003
*
arXiv
*
pre-print

Depending upon the method, the surviving non-supercompact

arXiv:math/0301016v1
fatcat:znxl2yyz4vcpjjzwbmyjvysrfm
*strongly*compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain*a**club*disjoint from the*set*of measurable cardinals ... We summarize the known methods of producing*a*non-supercompact*strongly*compact cardinal*and*describe some new variants. ... I to select*and*then play*a**condition*q 2α+1 above q 2α from the dense*set*D α . ...##
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Exactly controlling the non-supercompact strongly compact cardinals

2003
*
Journal of Symbolic Logic (JSL)
*

Depending upon the method, the surviving non-supercompact

doi:10.2178/jsl/1052669070
fatcat:szzk73eqxvbllne3j6v2notbpy
*strongly*compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain*a**club*disjoint from the*set*of measurable cardinals ... We summarize the known methods of producing*a*non-supercompact*strongly*compact cardinal*and*describe some new variants. ... I to select*and*then play*a**condition*q 2α+1 above q 2α from the dense*set*D α . ...##
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Completeness of the Gödel-Löb provability logic for the filter sequence of normal measures
[article]

2022
*
arXiv
*
pre-print

This result answers

arXiv:2110.02786v2
fatcat:wi33eqapsvajzhgctt5xcfal74
*a*question of Andreas Blass from 1990*and**a*related question of Beklemishev*and*Joosten. ... Assuming the existence of suitable large cardinals, we show it is consistent that the Provability logic 𝐆𝐋 is complete*with*respect to the filter sequence of normal measures. ... In fact, it is*adequate*to consider frames that are*finite*transitive irreflexive trees. ...
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