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Note on 𝖳𝖣 + 𝖣𝖢_ℝ implying 𝖠𝖣^L(ℝ)
[article]
2021
arXiv
pre-print
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A short core model induction proof of 𝖠𝖣^L(ℝ) 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]
2016
arXiv
pre-print
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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]
2019
arXiv
pre-print
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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)
2007
Bulletin of Symbolic Logic
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AbstractWe present a characterization of supercompactness measures for ω1in L(ℝ), 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(ℝ), and the second proving the uniqueness of the supercompactness measure overin L(ℝ) 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]
2012
arXiv
pre-print
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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]
2008
arXiv
pre-print
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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]
2021
arXiv
pre-print
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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
2009
Annals of Pure and Applied Logic
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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(A ⊕ A 0 )[f ],Ũ ). ...
doi:10.1016/j.apal.2008.12.001
fatcat:2j5s2npednedbh7e5jufbm35iy
Covering with universally Baire operators
2015
Advances in Mathematics
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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]
2016
arXiv
pre-print
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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
2015
Journal of Mathematical Logic
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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]
2022
arXiv
pre-print
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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]
2019
arXiv
pre-print
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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]
2022
arXiv
pre-print
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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
2017
Israel Journal of Mathematics
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2012 pre-print arXiv:1205.4275v1 fatcat:4cetveaplzcmtdc763silpin5qOther Versions
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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