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Equiconsistencies at subcompact cardinals

Itay Neeman, John Steel
2015 Archive for Mathematical Logic  
We present 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.  ... 
doi:10.1007/s00153-015-0465-4 fatcat:nlk4dcs2qjfuhg6hmk4yhuxjya

Hierarchies of forcing axioms I

Itay Neeman, Ernest Schimmerling
2008 Journal of Symbolic Logic (JSL)  
As a corollary, SPFA(ϲ-linked) and PFA(ϲ-linked) are each 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).  ... 
doi:10.2178/jsl/1208358756 fatcat:pc6unnma4ndmfpem3hqnga3jaa

Partial strong compactness and squares [article]

Yair Hayut
2018 arXiv   pre-print
In this paper we analyze the connection between some properties of partially strongly compact 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  ... 
arXiv:1804.05758v2 fatcat:c35yjf22sjds5bnyvyunxhj2da

Partial strong compactness and squares

Yair Hayut
2019 Fundamenta Mathematicae  
We analyze the connection between some properties of partially strongly compact 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}.  ... 
doi:10.4064/fm626-9-2018 fatcat:x3azwonlv5dwjpxy2w3kozavdm

Magidor-Malitz Reflection [article]

Yair Hayut
2018 arXiv   pre-print
We derive some combinatorial results and improve the known upper bound for the consistency of Chang's Conjecture 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.  ... 
arXiv:1512.09299v4 fatcat:h2iqg67efje5jbolmmgaendvju

Closure properties of measurable ultrapowers [article]

Philipp Lücke, Sandra Müller
2020 arXiv   pre-print
In one direction, a result of Sakai shows that, by collapsing a strongly compact 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.  ... 
arXiv:2009.09530v1 fatcat:ulnjnq43pvhrhj6tqwren5lnka

Some basic thoughts on the cofiality of Chang structures with an application to forcing [article]

Dominik Adolf
2020 arXiv   pre-print
Consider (κ^+++,κ^++) (κ^+,κ) where κ is an uncountable regular 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  ... 
arXiv:2003.11215v1 fatcat:wj63t2atcfbwffkynxmuztggla

Indexed squares

James Cummings, Ernest Schimmerling
2002 Israel Journal of Mathematics  
Jensen proved that a large 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.  ... 
doi:10.1007/bf02785851 fatcat:zxwn7e7x45hgxd6hyt77m72qxq

The Variety of Projection of a Tree-Prikry Forcing [article]

Tom Benhamou, Moti Gitik, Yair Hayut
2021 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 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.  ... 
arXiv:2109.09069v3 fatcat:5cudnsvvtnc55mdxzwi3zbftui

AN EQUICONSISTENCY RESULT ON PARTIAL SQUARES

JOHN KRUEGER, ERNEST SCHIMMERLING
2011 Journal of Mathematical Logic  
We prove that the following two statements are 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.  ... 
doi:10.1142/s0219061311000992 fatcat:vrkinndjvbbxffmayfotwt7pva

Hierarchies of forcing axioms II

Itay Neeman
2008 Journal of Symbolic Logic (JSL)  
A cardinal λ is , indescribable just in case that for every truth 〈Q, ψ〈 for λ, there exists &lt; 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.  ... 
doi:10.2178/jsl/1208359058 fatcat:lguneg7ypzdgvdseiso2weo2a4

Weak saturation properties and side conditions [article]

Monroe Eskew
2022 arXiv   pre-print
Towards combining "compactness" and "hugeness" properties 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.  ... 
arXiv:2209.00340v1 fatcat:um5p22bdtnhbdfgirttag2oegi

Simultaneous stationary reflection and square sequences [article]

Yair Hayut, Chris Lambie-Hanson
2017 arXiv   pre-print
The upper bound for the failure of κ for singular κ is a measurable subcompact cardinal. The notion of subcompact was defined by Jensen as a weakening of a supercompact cardinal.  ...  assumption below subcompact.  ... 
arXiv:1603.05556v2 fatcat:x2hdnq6zhraclklznzrfwqueny

Reflecting on Absolute Infinity

Philip Welch, Leon Horsten
2016 Journal of Philosophy  
of strong versions of GRP lie between the statement that postulates a 1extendible 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 + ).  ... 
doi:10.5840/jphil201611325 fatcat:sraoz3crabbw5pegvpugxdpcnq

The exact strength of generic absoluteness for the universally Baire sets [article]

Grigor Sargsyan, Nam Trang
2021 arXiv   pre-print
We show that over some mild large 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.  ... 
arXiv:2110.02725v1 fatcat:r7ifepdddvajno33o4l35cvmzm
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