The HOD Dichotomy

Woodin, W. Hugh; Davis, Jacob; Rodríguez, Daniel

doi:10.1017/cbo9781139208574.014

Cited by 9 publications

(15 citation statements)

References 3 publications

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“…Woodin speculates that the extension to the level of one supercompact cardinal should yield as a theorem that if δ is supercompact then there exists

N \subseteq HOD

such that N is a weak extender model for δ supercompact (cf. ). Theorem Suppose N is a weak extender model for δ supercompact and

γ > δ

is a singular cardinal.…”

Section: The Boldhod Hypothesismentioning

confidence: 97%

“…Proposition (1)If

double-struckP

is a weakly homogeneous and ordinal definable poset in

boldV

and G is a

boldV

‐generic filter on

double-struckP

, then

{boldHOD}^{V [G]} \subseteq {boldHOD}^{boldV}

(Folklore; ). (2)If

κ > ω

is an regular cardinal,

double-struckP

is a poset with

| P | < κ, G

is a

double-struckP

‐generic over

boldV

, then in

V [G], V

is Σ 2 definable from

V \cap ℘ (κ)

[, Lemma 4, Theorem 5]. (3)For every ordinal κ, there exists

B \in HOD

such that

HOD ⊧ B

is a a complete Boolean algebra, and for any

E \subseteq κ

, there exists a

boldHOD

‐generic filter G on

double-struckB

such that

HOD false[E false] \subseteq {boldHOD}_{{G}} = {boldHOD}_{{E}} = HOD false[G false]

(Vopěnka, cf. [, Theorem 6]). (4)Let G be generic on

double-struckB

. If M is a model of

sans-serifZFC

such that

V \subseteq M \subseteq V [G]

, then there exists a com...…”

Section: The Boldhod Hypothesismentioning

confidence: 99%

“…Suppose G is a

boldV

‐generic filter on

double-struckP

. Then

{HOD}^{V}

is a generic extension of

{HOD}^{boldV false[G false]}

[, Corollary 7].…”

Section: The Boldhod Hypothesismentioning

confidence: 99%

“…In Woodin's recent paper , the

boldHOD

Conjecture denotes the following statement as in Definition . In Woodin's old paper such as , the

boldHOD

Conjecture denotes the same statement as the

boldHOD

Hypothesis. Definition The

boldHOD

Conjecture denotes the following statement: the theory

ZFC +

“there exists a supercompact cardinal” proves the

boldHOD

Hypothesis.…”

Section: The Boldhod Hypothesismentioning

confidence: 99%

“…The following Theorems & are a reformulation and summary of Woodin's results in (e.g., [, Theorem 212], [, Theorem 19], etc.). Theorem Suppose δ is

boldHOD

‐supercompact .…”

Section: The Boldhod Hypothesismentioning

confidence: 99%

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The Hypothesis and a supercompact cardinal

Cheng

2017

Mathematical Logic Qtrly

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In this paper, we prove that: if κ is supercompact and the boldHOD Hypothesis holds, then there is a proper class of regular cardinals in Vκ which are measurable in boldHOD. Woodin also proved this result independently . As a corollary, we prove Woodin's Local Universality Theorem. This work shows that under the assumption of the boldHOD Hypothesis and supercompact cardinals, large cardinals in boldV are reflected to be large cardinals in boldHOD in a local way, and reveals the huge difference between boldHOD‐supercompact cardinals and supercompact cardinals under the boldHOD Hypothesis.

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“…Woodin speculates that the extension to the level of one supercompact cardinal should yield as a theorem that if δ is supercompact then there exists

N \subseteq HOD

such that N is a weak extender model for δ supercompact (cf. ). Theorem Suppose N is a weak extender model for δ supercompact and

γ > δ

is a singular cardinal.…”

Section: The Boldhod Hypothesismentioning

confidence: 97%

“…Proposition (1)If

double-struckP

is a weakly homogeneous and ordinal definable poset in

boldV

and G is a

boldV

‐generic filter on

double-struckP

, then

{boldHOD}^{V [G]} \subseteq {boldHOD}^{boldV}

(Folklore; ). (2)If

κ > ω

is an regular cardinal,

double-struckP

is a poset with

| P | < κ, G

is a

double-struckP

‐generic over

boldV

, then in

V [G], V

is Σ 2 definable from

V \cap ℘ (κ)

[, Lemma 4, Theorem 5]. (3)For every ordinal κ, there exists

B \in HOD

such that

HOD ⊧ B

is a a complete Boolean algebra, and for any

E \subseteq κ

, there exists a

boldHOD

‐generic filter G on

double-struckB

such that

HOD false[E false] \subseteq {boldHOD}_{{G}} = {boldHOD}_{{E}} = HOD false[G false]

(Vopěnka, cf. [, Theorem 6]). (4)Let G be generic on

double-struckB

. If M is a model of

sans-serifZFC

such that

V \subseteq M \subseteq V [G]

, then there exists a com...…”

Section: The Boldhod Hypothesismentioning

confidence: 99%

“…Suppose G is a

boldV

‐generic filter on

double-struckP

. Then

{HOD}^{V}

is a generic extension of

{HOD}^{boldV false[G false]}

[, Corollary 7].…”

Section: The Boldhod Hypothesismentioning

confidence: 99%

“…In Woodin's recent paper , the

boldHOD

Conjecture denotes the following statement as in Definition . In Woodin's old paper such as , the

boldHOD

Conjecture denotes the same statement as the

boldHOD

Hypothesis. Definition The

boldHOD

Conjecture denotes the following statement: the theory

ZFC +

“there exists a supercompact cardinal” proves the

boldHOD

Hypothesis.…”

Section: The Boldhod Hypothesismentioning

confidence: 99%

“…The following Theorems & are a reformulation and summary of Woodin's results in (e.g., [, Theorem 212], [, Theorem 19], etc.). Theorem Suppose δ is

boldHOD

‐supercompact .…”

Section: The Boldhod Hypothesismentioning

confidence: 99%

See 3 more Smart Citations

The Hypothesis and a supercompact cardinal

Cheng

2017

Mathematical Logic Qtrly

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Choiceless Löwenheim–Skolem Property and Uniform Definability of Grounds

Usuba

2021

Springer Proceedings in Mathematics &Amp; Statistics

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Extendible cardinals and the mantle

Usuba

2018

Arch. Math. Logic

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The HOD Dichotomy

Cited by 9 publications

References 3 publications

The Hypothesis and a supercompact cardinal

The Hypothesis and a supercompact cardinal

Choiceless Löwenheim–Skolem Property and Uniform Definability of Grounds

Extendible cardinals and the mantle

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