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

2021
*
arXiv
*
pre-print

*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. ...

##
###
Scales in hybrid mice over R
[article]

2016
*
arXiv
*
pre-print

We develop

arXiv:1210.7258v4
fatcat:5g4vsvbuwbbhzhusvsyewlsmkq
*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] . ...##
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Definable Hamel bases and AC_ω(R)
[article]

2019
*
arXiv
*
pre-print

There is

arXiv:1901.04750v2
fatcat:v6oqbwrbsrd2de24booydqgwiq
*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. ...##
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Inner Models and Ultrafilters In L(ℝ)

2007
*
Bulletin of Symbolic Logic
*

AbstractWe present

doi:10.2178/bsl/1174668217
fatcat:jpmokosadnellhy26yucuqrlwi
*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*. ...##
###
The core model induction beyond L(R): non-tame mouse from PFA
[article]

2012
*
arXiv
*
pre-print

We show that PFA implies that there is

arXiv:1206.2714v1
fatcat:5tr6us32g5fcvnzaayxmvq7624
*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. ...##
###
Sobolev W^1_p-spaces on closed subsets of R^n
[article]

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.) ...

##
###
Strongly compact cardinals and ordinal definability
[article]

2021
*
arXiv
*
pre-print

We show that the

arXiv:2107.00513v1
fatcat:dg2mbfbfgjhgjcsanzu2b7r6wm
*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. ...##
###
The strength of choiceless patterns of singular and weakly compact cardinals

2009
*
Annals of Pure and Applied Logic
*

in the L(

doi:10.1016/j.apal.2008.12.001
fatcat:2j5s2npednedbh7e5jufbm35iy
*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 ). ...##
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Covering with universally Baire operators

2015
*
Advances in Mathematics
*

We introduce

doi:10.1016/j.aim.2014.09.016
fatcat:rx7ftuuumfgxpaokrnvuionng4
*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 ...##
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The HOD Dichotomy
[article]

2016
*
arXiv
*
pre-print

We define the

arXiv:1605.00613v1
fatcat:fgt2m3qo7recbhkur4fowk5tiy
*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. ...##
###
Collapsing the cardinals of HOD

2015
*
Journal of Mathematical Logic
*

In particular the rank-initial segment W_κ is

doi:10.1142/s0219061315500075
fatcat:mufqioqwmfabzjgret4ghsrc34
*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. ...##
###
Projective Completion of Moduli of t-Connections on Curves in Positive and Mixed Characteristic
[article]

2022
*
arXiv
*
pre-print

We generalize

arXiv:2104.12209v2
fatcat:o7ldrgdwozffjn2wi4hashm46q
*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. ...##
###
Inner mantles and iterated HOD
[article]

2019
*
arXiv
*
pre-print

We present

arXiv:1810.08702v2
fatcat:guqueg35ljdzrdelsz5xfk7bja
*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*R*α ,*R*α*Ǎ*dd(α,λ α ) =*Ȧ*dd(α,λ α ) V*R*α , and*R*α V ⊆ V*R*α satisfies the α-*cover*property*for*subsets*of*λ α (that is, any*A*⊂ λ α from V*R*α*of*size <α in that ...##
###
P=W conjectures for character varieties with symplectic resolution
[article]

2022
*
arXiv
*
pre-print

To this end, we prove auxiliary results

arXiv:2006.08752v4
fatcat:pkkfrhqdsjdpvjbk27utanc5gy
*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. ...##
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Square principles in ℙmax extensions

2017
*
Israel Journal of Mathematics
*

By forcing with P_ max over strong models

doi:10.1007/s11856-017-1444-8
fatcat:m47x56z3evfj5dtlzqy6wxy3oe
*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*). ...
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