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Equiconsistencies at subcompact cardinals

2015
*
Archive for Mathematical Logic
*

We present

doi:10.1007/s00153-015-0465-4
fatcat:nlk4dcs2qjfuhg6hmk4yhuxjya
*equiconsistency*results*at*the level of*subcompact**cardinals*. ... We present*equiconsistency*results*at*the level of*subcompact**cardinals*. The methods we use extend further, to levels which are interlaced with the axioms κ is κ +(n) supercompact, for n < ω. ... Suppose that δ is a Woodin*cardinal*, δ is threadable, and δ + is threadable. Then δ is Π 2 1*subcompact*in a class inner model. Corollary 1.6. The following are*equiconsistent*: 1. ...##
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Hierarchies of forcing axioms I

2008
*
Journal of Symbolic Logic (JSL)
*

As a corollary, SPFA(ϲ-linked) and PFA(ϲ-linked) are each

doi:10.2178/jsl/1208358756
fatcat:pc6unnma4ndmfpem3hqnga3jaa
*equiconsistent*with the existence of a -indescribable*cardinal*. ... Our results are in terms of (θ, Γ)-*subcompactness*, which is a new large*cardinal*notion that combines the ideas behind*subcompactness*and Γ-indescribability. ... Conjecture 14 . 14 The theories ZFC + PFA(c + -linked) and ZFC + There is a*cardinal*λ that is (λ + , Σ 2 1 )-*subcompact*are*equiconsistent*. Corollary 16 . 16 SPFA(c + -linked) implies MM(c). ...##
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Partial strong compactness and squares
[article]

2018
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arXiv
*
pre-print

In this paper we analyze the connection between some properties of partially strongly compact

arXiv:1804.05758v2
fatcat:c35yjf22sjds5bnyvyunxhj2da
*cardinals*: the completion of filters of certain size and instances of the compactness of L_κ,κ. ... The existence of κ which is κ-compact is*equiconsistent*with the existence of a*cardinal*δ which is δ + -Π 1 1 -*subcompact*. ... The least large*cardinal*which implies the failure of squares*at*its successor seems to be*subcompactness*(see [12] ), and the results of this paper suggest that κ-compactness is related to a strong version ...##
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Partial strong compactness and squares

2019
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Fundamenta Mathematicae
*

We analyze the connection between some properties of partially strongly compact

doi:10.4064/fm626-9-2018
fatcat:x3azwonlv5dwjpxy2w3kozavdm
*cardinals*: the completion of filters of certain size and instances of the compactness of Lκ,κ. ... We prefer the more cumbersome name in order to avoid inconsistency with the term κ-compact, which refers to a*cardinal*κ that has the κ-filter extension property. ... The existence of κ which is κ-compact is*equiconsistent*with the existence of a*cardinal*δ which is δ + -Π 1 1 -*subcompact*. For a set A ⊆ λ, let I(A) = { X, Z | A ∩ X ∈ Z}. ...##
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Magidor-Malitz Reflection
[article]

2018
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arXiv
*
pre-print

We derive some combinatorial results and improve the known upper bound for the consistency of Chang's Conjecture

arXiv:1512.09299v4
fatcat:h2iqg67efje5jbolmmgaendvju
*at*successor of singular*cardinals*. ... A*cardinal*κ is*subcompact*if it is (+1)-*subcompact*. ... We remark that for successor of a regular*cardinal*κ, the existence of such an ideal I is*equiconsistent*with the existence of a measurable*cardinal*. ...##
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Closure properties of measurable ultrapowers
[article]

2020
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arXiv
*
pre-print

In one direction, a result of Sakai shows that, by collapsing a strongly compact

arXiv:2009.09530v1
fatcat:ulnjnq43pvhrhj6tqwren5lnka
*cardinal*to become the double successor of a measurable*cardinal*, it is possible to obtain a model of set theory in which ... The following statements are*equiconsistent*over the theory ZFC: (i) There exists a (δ + 2)-strong*cardinal*δ. ... Assume that V is a Jensen-style extender model that does not have a*subcompact**cardinal*. ...##
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Some basic thoughts on the cofiality of Chang structures with an application to forcing
[article]

2020
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arXiv
*
pre-print

Consider (κ^+++,κ^++) (κ^+,κ) where κ is an uncountable regular

arXiv:2003.11215v1
fatcat:wj63t2atcfbwffkynxmuztggla
*cardinal*. By a result of Shelah's we have cof(X ∩κ^++) = κ for almost all X ⊂κ^+++ witnessing this. ... Its most basic form (often known simply as the Chang conjecture), (ℵ 2 , ℵ 1 ) (ℵ 1 , ℵ 0 ) in our notation, is*equiconsistent*with an ω 1 -Erdős*cardinal*. ... We will use these results together with arguments from [EH18] to prove this: Theorem 1.1: Let κ < λ < δ be three*cardinals*such that κ is λ-supercompact, λ is +2-*subcompact*, and δ is Woodin or strongly ...##
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Indexed squares

2002
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Israel Journal of Mathematics
*

Jensen proved that a large

doi:10.1007/bf02785851
fatcat:zxwn7e7x45hgxd6hyt77m72qxq
*cardinal*property slightly stronger than 1extendibility is incompatible with square; we prove this is close to optimal by showing that 1-extendibility is compatible with square ...*Subcompactness*is a new large*cardinal*property that was introduced by Jensen;*subcompactness*follows from supercompactness. ... Jensen's argument shows that if κ is*subcompact*then κ fails. We note that a*subcompact**cardinal*need not be measurable. ...##
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The Variety of Projection of a Tree-Prikry Forcing
[article]

2021
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arXiv
*
pre-print

We study which κ-distributive forcing notions of size κ can be embedded into tree Prikry forcing notions with κ-complete ultrafilters under various large

arXiv:2109.09069v3
fatcat:5cudnsvvtnc55mdxzwi3zbftui
*cardinal*assumptions. ... The failure of square*at*two consecutive*cardinals*seem to have very high consistency strength, which made the conjecture that κ-compactness is*equiconsistent*with Π 1 1 -*subcompactness*plausible. ... This distinction leads to the realm of*subcompact**cardinals*.*Subcompact**cardinals*were defined by R. Jensen: Definition 27. ...##
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AN EQUICONSISTENCY RESULT ON PARTIAL SQUARES

2011
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Journal of Mathematical Logic
*

We prove that the following two statements are

doi:10.1142/s0219061311000992
fatcat:vrkinndjvbbxffmayfotwt7pva
*equiconsistent*: there exists a greatly Mahlo*cardinal*; there exists a regular uncountable*cardinal*κ such that no stationary subset of κ + ∩ cof(κ) carries ... A Related*Equiconsistency*Result Let λ be a regular uncountable*cardinal*. ... A famous theorem in set theory is the result that the failure of the square principle κ , for a regular uncountable*cardinal*κ, is*equiconsistent*with a Mahlo*cardinal*. ...##
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Hierarchies of forcing axioms II

2008
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Journal of Symbolic Logic (JSL)
*

A

doi:10.2178/jsl/1208359058
fatcat:lguneg7ypzdgvdseiso2weo2a4
*cardinal*λ is , indescribable just in case that for every truth 〈Q, ψ〈 for λ, there exists < λ so that is a*cardinal*and 〈Q ∩ , ψ) is a truth for . ... More generally, an interval of*cardinals*[κ, λ] with κ ≤ λ is indescribable if for every truth 〈Q, ψ〈 for λ, there exists , and π: → H λ so that is a*cardinal*, is a truth for , and π is elementary from ... We also say that κ is (λ, Σ 2 1 )-*subcompact*in this case.*At*the lowest end, [κ, κ] is Σ 2 1 indescribable just in case that κ is Σ 2 1 indescribable. ...##
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Weak saturation properties and side conditions
[article]

2022
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arXiv
*
pre-print

Towards combining "compactness" and "hugeness" properties

arXiv:2209.00340v1
fatcat:um5p22bdtnhbdfgirttag2oegi
*at*ω_2, we use Neeman's side-conditions forcing to reduce the upper bound on the consistency of the weak Chang's Conjecture*at*ω_2 and show it ... "strong ideal") is*equiconsistent*with a Woodin*cardinal*. ... Along the way, we reduce the upper bound on the consistency strength of wCC*at*ω 2 down from almost-hugeness to (+2)-*subcompactness*. ...##
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Simultaneous stationary reflection and square sequences
[article]

2017
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arXiv
*
pre-print

The upper bound for the failure of κ for singular κ is a measurable

arXiv:1603.05556v2
fatcat:x2hdnq6zhraclklznzrfwqueny
*subcompact**cardinal*. The notion of*subcompact*was defined by Jensen as a weakening of a supercompact*cardinal*. ... assumption below*subcompact*. ...##
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Reflecting on Absolute Infinity

2016
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Journal of Philosophy
*

of strong versions of GRP lie between the statement that postulates a 1extendible

doi:10.5840/jphil201611325
fatcat:sraoz3crabbw5pegvpugxdpcnq
*cardinal*and the statement that postulates the existence of the*cardinal*motivating the GRP: a*subcompact**cardinal*. 50 ... For all weaker large*cardinal*axioms (with critical point κ), the embeddings that they postulate are continuous*at*κ + , in the sense that sup j"κ + = j(κ + ). ...##
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The exact strength of generic absoluteness for the universally Baire sets
[article]

2021
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arXiv
*
pre-print

We show that over some mild large

arXiv:2110.02725v1
fatcat:r7ifepdddvajno33o4l35cvmzm
*cardinal*theory, Sealing is*equiconsistent*with LSA-over-uB. In fact, we isolate an exact large*cardinal*theory that is*equiconsistent*with both (see dfn:hod_pm). ... A variation of Sealing, called Tower Sealing, is also shown to be*equiconsistent*with Sealing over the same large*cardinal*theory. ... If in addition there is no inner model with a*subcompact**cardinal*then M ν . With more work, the conjecture can also be stated without assuming the large*cardinals*. ...
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