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Note on 𝖳𝖣 + 𝖣𝖢_ℝ implying 𝖠𝖣^L(ℝ) [article]

Sean Cody
2021 arXiv   pre-print
A short core model induction proof of 𝖠𝖣^L() from 𝖳𝖣 + 𝖣𝖢_ℝ.  ...  By cheapo covering and the fact that L J [x] is a size < ω V 1 forcing extension of HOD L J [x] a , we can choose some λ < ω V 1 such that λ +K J = λ +L J [x] Let g ∈ V be Col(ω, λ)-generic over L J [x  ...  HOD L[S,x] S |= ω L[S,x] 1 is measurable for a cone of x.  ... 
arXiv:2110.01135v1 fatcat:djmlifpvpjb3xfrom3kmw2ldzi

Scales in hybrid mice over R [article]

Farmer Schlutzenberg, Nam Trang
2016 arXiv   pre-print
We develop a general theory of strategic mice, prove their condensation properties, and analyze the scales pattern in the stack of Θ-g-organized F-mice over R, Lp^GF(R), for a class of nice operators F  ...  The article also uses a small part of the theory (and notation) of hod mice, as covered in the first parts of [5] .  ...  We assume only a basic knowledge of hod mice; more than enough is covered in the first sections of [5] .  ... 
arXiv:1210.7258v4 fatcat:5g4vsvbuwbbhzhusvsyewlsmkq

Definable Hamel bases and AC_ω(R) [article]

Vladimir Kanovei, Ralf Schindler
2019 arXiv   pre-print
There is a model of ZF with a Δ^1_3 definable Hamel basis in which AC_ω(R) fails.  ...  We may force with the forcings from [3] and [1] to add a Burstin basis and a Mazurkiewicz set, respectively, over H (without adding any reals), but then those sets won't be definable in that extension  ...  The model H = H(L) = HOD L[g] A∪{A} Lemma 2. 1 1 Inside H, A is a (lightface) Π 1 2 set. Proof. Let ϕ(−) be the Π 1 2 formula from the proof of Lemma 1.8.  ... 
arXiv:1901.04750v2 fatcat:v6oqbwrbsrd2de24booydqgwiq

Inner Models and Ultrafilters In L(ℝ)

Itay Neeman
2007 Bulletin of Symbolic Logic  
AbstractWe present a characterization of supercompactness measures for ω1in L(), and of countable products of such measures, using inner models.  ...  We give two applications of this characterization, the first obtaining the consistency ofwith ZFC+ADL(), and the second proving the uniqueness of the supercompactness measure overin L() for.  ...  M k ∞ is an initial segment of HOD L(R) (u). Moreover there are ordinals ν k , k < ω, independent of M and cofinal in Θ, so that ON ∩ M k ∞ ≥ ν k for each k.  ... 
doi:10.2178/bsl/1174668217 fatcat:jpmokosadnellhy26yucuqrlwi

The core model induction beyond L(R): non-tame mouse from PFA [article]

Grigor Sargsyan
2012 arXiv   pre-print
We show that PFA implies that there is a non-tame mouse.  ...  The proof of this lemma is essentially the covering argument. The following lemma is our main tool for extending iteration strategies to κ + -iteration strategies.  ...  Then W κ,F,g (a) W κ,F (a), K κ,F,g (a) K κ,F (a) and Lp F,g (a) Lp F (a). Definition 3.3 (Covering by Lp, lpc) Suppose κ is an uncountable cardinal.  ... 
arXiv:1206.2714v1 fatcat:5tr6us32g5fcvnzaayxmvq7624

Sobolev W^1_p-spaces on closed subsets of R^n [article]

Pavel Shvartsman
2008 arXiv   pre-print
For each p>n we use local oscillations and doubling measures to give intrinsic characterizations of the restriction of the Sobolev space W_p^1(R^n) to an arbitrary closed subset of R^n.  ...  Then for every G ∈ L β (R n ) the following inequality { Q∈W(A) |Q| |G γQ | β } 1 β ≤ C G L β (R n ) . hods. Here C is a constant depending only on n and β.  ...  (Recall that W (S) is a covering of R n \ S by non-overlapping cubes such that diam Q k ∼ dist(Q k , S), k ∈ I.)  ... 
arXiv:0806.2430v1 fatcat:7zd3kjwvanctdack2kglg6ir3u

Strongly compact cardinals and ordinal definability [article]

Gabriel Goldberg
2021 arXiv   pre-print
We show that the HOD hypothesis is equivalent to a uniqueness property of elementary embeddings of levels of the cumulative hierarchy.  ...  We show that assuming there is a strongly compact cardinal and the HOD hypothesis holds, there is no elementary embedding from HOD to HOD, settling a question of Woodin.  ...  By Lemma 2.7, HOD has the (ω 1 , κ)-cover property. Let i : HOD → N be given by the extender of length κ derived from j, and let k : N → HOD be the factor embedding, so crit(k) ≥ κ.  ... 
arXiv:2107.00513v1 fatcat:dg2mbfbfgjhgjcsanzu2b7r6wm

The strength of choiceless patterns of singular and weakly compact cardinals

Daniel Busche, Ralf Schindler
2009 Annals of Pure and Applied Logic  
in the L(R) of a generic extension of HOD: (a) ZF + every uncountable cardinal is singular, and (b) ZF + every infinite successor cardinal is weakly compact and every uncountable limit cardinal is singular  ...  We extend the core model induction technique to a choiceless context, and we exploit it to show that each one of the following two hypotheses individually implies that AD, the Axiom of Determinacy, holds  ...  δ But this contradicts "weak covering for K(A)" which implies cof(δ +K (A) ) ≥ δ in ult(R(AA 0 )[f ],Ũ ).  ... 
doi:10.1016/j.apal.2008.12.001 fatcat:2j5s2npednedbh7e5jufbm35iy

Covering with universally Baire operators

Grigor Sargsyan
2015 Advances in Mathematics  
We introduce a covering conjecture and show that it holds below AD R + "Θ is regular".  ...  This theory generalizes to K c,F . We refer the reader to Section 1.3 of [12] for more details.  ...  I arrived at the conjecture by trying to understand how core model We can then define m : M η → j(P) by m(x) = y if and only if whenever R ∈ pI(Q, Ψ) is such that for some z ∈ R, π Ψ R,∞,j(η) (z) = x then  ... 
doi:10.1016/j.aim.2014.09.016 fatcat:rx7ftuuumfgxpaokrnvuionng4

The HOD Dichotomy [article]

Hugh Woodin, Jacob Davis, Daniel Rodriguez
2016 arXiv   pre-print
We define the HOD conjecture, prove it is equivalent to a formulation in terms of weak extender models for supercompactness, and give some of its consequences.  ...  This paper provides an accessible introduction to some of the work of Woodin on suitable extender models.  ...  Lemma 16. If N is a weak extender model for δ supercompact, then it has the δ-covering property. Proof. Note that it is enough to prove δ-covering for sets of ordinals.  ... 
arXiv:1605.00613v1 fatcat:fgt2m3qo7recbhkur4fowk5tiy

Collapsing the cardinals of HOD

James Cummings, Sy David Friedman, Mohammad Golshani
2015 Journal of Mathematical Logic  
In particular the rank-initial segment W_κ is a model of ZFC in which (α^+)^HOD < α^+ for every infinite cardinal α.  ...  Assuming that GCH holds and κ is κ^+3-supercompact, we construct a generic extension W of V in which κ remains strongly inaccessible and (α^+)^HOD < α^+ for every infinite cardinal α < κ.  ...  It follows that a failure of weak covering over HOD will also be a failure of weak covering over every COLLAPSING THE CARDINALS OF HOD 3 reasonably definable inner model.  ... 
doi:10.1142/s0219061315500075 fatcat:mufqioqwmfabzjgret4ghsrc34

Projective Completion of Moduli of t-Connections on Curves in Positive and Mixed Characteristic [article]

Mark Andrea A. de Cataldo, Siqing Zhang
2022 arXiv   pre-print
We generalize a compactification technique due to C. Simpson in the context of 𝔾_m-actions over the ground field of complex numbers, to the case of a universally Japanese base ring.  ...  We apply these projectivity results to the Hodge, de Rham, and Dolbeault moduli spaces for curves, with special regards to ground fields of positive characteristic.  ...  We thank the referee for the very good suggestions.  ... 
arXiv:2104.12209v2 fatcat:o7ldrgdwozffjn2wi4hashm46q

Inner mantles and iterated HOD [article]

Jonas Reitz, Kameryn J Williams
2019 arXiv   pre-print
We present a class forcing notion M(η), uniformly definable for ordinals η, which forces the ground model to be the η-th inner mantle of the extension, in which the sequence of inner mantles has length  ...  This answers a conjecture of Fuchs, Hamkins, and Reitz [FHR15] in the positive.  ...  of P α (3) R α Ǎ dd(α, 1) = Ȧ dd(α, 1) V Rα , R α Ǎ dd(α,λ α ) = Ȧ dd(α,λ α ) V Rα , and R α V ⊆ V Rα satisfies the α-cover property for subsets of λ α (that is, any A ⊂ λ α from V Rα of size <α in that  ... 
arXiv:1810.08702v2 fatcat:guqueg35ljdzrdelsz5xfk7bja

P=W conjectures for character varieties with symplectic resolution [article]

Camilla Felisetti, Mirko Mauri
2022 arXiv   pre-print
To this end, we prove auxiliary results of independent interest, like the construction of a relative compactification of the Hodge moduli space for reductive algebraic groups, and the projectivity of the  ...  We establish P=W and PI=WI conjectures for character varieties with structural group GL_n and SL_n which admit a symplectic resolution, i.e. for genus 1 and arbitrary rank, and genus 2 and rank 2.  ...  Let f Hod : R(M Hod (X, G)) → M Hod (X, G) be the functorial resolution of M Hod (X, G), equivalently in the analytic or algebraic category by Lemma 3.5.  ... 
arXiv:2006.08752v4 fatcat:pkkfrhqdsjdpvjbk27utanc5gy

Square principles in ℙmax extensions

Andrés Eduardo Caicedo, Paul Larson, Grigor Sargsyan, Ralf Schindler, John Steel, Martin Zeman
2017 Israel Journal of Mathematics  
By forcing with P_ max over strong models of determinacy, we obtain models where different square principles at ω_2 and ω_3 fail. In particular, we obtain a model of 2^_0=2^_1=_2 + (ω_2) + (ω_3).  ...  Woodin [Woo10, §3.1] presents a series of covering theorems for P max extensions.  ...  Using now the Covering Theorem 4.2, we see that |C ∩ α| ≤ ℵ 1 , so there is a D ∈ V with C ∩ α ⊆ D and |D| ≤ ℵ 1 in V. By the Coding Lemma, D ∈ L(A, R) and |D| ≤ ℵ 1 in L(A, R).  ... 
doi:10.1007/s11856-017-1444-8 fatcat:m47x56z3evfj5dtlzqy6wxy3oe
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