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Strongly adequate sets and adding a club with finite conditions

John Krueger
2013 Archive for Mathematical Logic  
We present 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.  ... 
doi:10.1007/s00153-013-0358-3 fatcat:qilunpmbwnakzosvjwmlfwkbca

Adding a club with finite conditions, Part II

John Krueger
2014 Archive for Mathematical Logic  
We define 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.  ... 
doi:10.1007/s00153-014-0406-7 fatcat:bq3ecooe45etjpyjmnnlml2kl4

Coherent adequate forcing and preserving CH

John Krueger, Miguel Angel Mota
2015 Journal of Mathematical Logic  
We show that there exists 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] .  ... 
doi:10.1142/s0219061315500051 fatcat:m3lvu6bfbferxiokyanazdt5ki

Coherent Adequate Forcing and Preserving CH [article]

John Krueger, Miguel Angel Mota
2014 arXiv   pre-print
We show that there exists 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] .  ... 
arXiv:1406.3302v1 fatcat:tjmvubowwvh6rcf7ctxlhl42fe

QUOTIENTS OF STRONGLY PROPER FORCINGS AND GUESSING MODELS

SEAN COX, JOHN KRUEGER
2016 Journal of Symbolic Logic (JSL)  
Specifically, we prove that quotients of forcing posets which have universal 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]).  ... 
doi:10.1017/jsl.2015.46 fatcat:jmb7i4u5rnfp3luuavikvadg2q

Quotients of Strongly Proper Forcings and Guessing Models [article]

Sean Cox, John Krueger
2015 arXiv   pre-print
Specifically, we prove that quotients of forcing posets which have simple universal 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]).  ... 
arXiv:1406.3306v2 fatcat:jky47xnjpje27bb2pam7c6sbji

Coherent Adequate Sets and Forcing Square [article]

John Krueger
2014 arXiv   pre-print
We introduce the idea of 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.  ... 
arXiv:1406.3309v1 fatcat:oh6q7ibyh5hxxalr5qbjcg53u4

Coherent adequate sets and forcing square

John Krueger
2014 Fundamenta Mathematicae  
We introduce the idea of 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.  ... 
doi:10.4064/fm224-3-5 fatcat:dn5k5bc6vjce3bnjlhdpt2yhke

Forcing with Adequate Sets of Models as Side Conditions [article]

John Krueger
2016 arXiv   pre-print
We present 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.  ... 
arXiv:1406.3307v3 fatcat:e7n2s3vv6nftdlmx6a6szwl23e

Mitchell's Theorem Revisited [article]

Thomas Gilton, John Krueger
2016 arXiv   pre-print
In this paper we give 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.  ... 
arXiv:1506.01983v3 fatcat:v4aawtn5oza3topqpbs3zj4quq

Proper forcing remastered [article]

Boban Velickovic, Giorgio Venturi
2013 arXiv   pre-print
We then discuss some applications: the Friedman-Mitchell poset for 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.  ... 
arXiv:1110.0610v3 fatcat:agafxndjjbhxvnlw7h64urpg2i

The Approachability Ideal Without a Maximal Set [article]

John Krueger
2018 arXiv   pre-print
We develop 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.  ... 
arXiv:1607.04772v3 fatcat:nmekousx3bdi3nxe2lw6yqblpa

Exactly controlling the non-supercompact strongly compact cardinals [article]

Arthur W. Apter, Joel David Hamkins
2003 arXiv   pre-print
Depending upon the method, the surviving non-supercompact 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 .  ... 
arXiv:math/0301016v1 fatcat:znxl2yyz4vcpjjzwbmyjvysrfm

Exactly controlling the non-supercompact strongly compact cardinals

Arthur W. Apter, Joel David Hamkins
2003 Journal of Symbolic Logic (JSL)  
Depending upon the method, the surviving non-supercompact 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 .  ... 
doi:10.2178/jsl/1052669070 fatcat:szzk73eqxvbllne3j6v2notbpy

Completeness of the Gödel-Löb provability logic for the filter sequence of normal measures [article]

Mohammad Golshani, Reihane Zoghifard
2022 arXiv   pre-print
This result answers 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.  ... 
arXiv:2110.02786v2 fatcat:wi33eqapsvajzhgctt5xcfal74
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