Ultimate-L
Definition
Suppose λ is an uncountable cardinal.
I λ is a singular cardinal if there exists a cofinal set X ? λ
such that |X| < λ.
I λ is a regular cardinal if there does not exist a cofinal set
X ? λ such that |X| < λ.
Definition
Suppose λ is an uncountable cardinal.
I λ is a singular cardinal if there exists a cofinal set X ? λ
such that |X| < λ.
I λ is a regular cardinal if there does not exist a cofinal set
X ? λ such that |X| < λ.
Lemma (Axiom of Choice)
Every (infinite) successor cardinal is a regular cardinal.
Definition
Suppose λ is an uncountable cardinal.
I λ is a singular cardinal if there exists a cofinal set X ? λ
such that |X| < λ.
I λ is a regular cardinal if there does not exist a cofinal set
X ? λ such that |X| < λ.
Lemma (Axiom of Choice)
Every (infinite) successor cardinal is a regular cardinal.
Definition
Suppose λ is an uncountable cardinal. Then cof(λ) is the
minimum possible |X| where X ? λ is cofinal in λ.
I cof(λ) is always a regular cardinal.
I If λ is regular then cof(λ) = λ.
I If λ is singular then cof(λ) < λ.
The Jensen Dichotomy Theorem
Theorem (Jensen)
Exactly one of the following holds.
(1) For all singular cardinals γ, γ is a singular cardinal in L and
γ
+ = (γ
+)
L
.
I L is close to V.
(2) Every uncountable cardinal is a regular limit cardinal in L.
I L is far from V.
The Jensen Dichotomy Theorem
Theorem (Jensen)
Exactly one of the following holds.
(1) For all singular cardinals γ, γ is a singular cardinal in L and
γ
+ = (γ
+)
L
.
I L is close to V.
(2) Every uncountable cardinal is a regular limit cardinal in L.
I L is far from V.
A strong version of Scott’s Theorem:
Theorem (Silver)
Assume that there is a measurable cardinal.
I Then L is far from V.
Tarski’s Theorem and G¨odel’s Response
Theorem (Tarski)
Suppose M |= ZF and let X be the set of all a ∈ M such that a is
definable in M without parameters.
I Then X is not definable in M without parameters.
Tarski’s Theorem and G¨odel’s Response
Theorem (Tarski)
Suppose M |= ZF and let X be the set of all a ∈ M such that a is
definable in M without parameters.
I Then X is not definable in M without parameters.
Theorem (G¨odel)
Suppose that M |= ZF and let X be the set of all a ∈ M such that
a is definable in M from b for some ordinal b of M.
I Then X is Σ2-definable in M without parameters.
G¨odel’s transitive class HOD
I Recall that a set M is transitive if every element of M is a
subset of M.
Definition
HOD is the class of all sets X such that there exist α ∈ Ord and
M ? Vα such that
1. X ∈ M and M is transitive.
2. Every element of M is definable in Vα from ordinal
parameters.
G¨odel’s transitive class HOD
I Recall that a set M is transitive if every element of M is a
subset of M.
Definition
HOD is the class of all sets X such that there exist α ∈ Ord and
M ? Vα such that
1. X ∈ M and M is transitive.
2. Every element of M is definable in Vα from ordinal
parameters.
I (ZF) The Axiom of Choice holds in HOD.
G¨odel’s transitive class HOD
I Recall that a set M is transitive if every element of M is a
subset of M.
Definition
HOD is the class of all sets X such that there exist α ∈ Ord and
M ? Vα such that
1. X ∈ M and M is transitive.
2. Every element of M is definable in Vα from ordinal
parameters.
I (ZF) The Axiom of Choice holds in HOD.
I L ? HOD.
I HOD is the union of all transitive sets M such that every
element of M is definable in V from ordinal parameters.
I By G¨odel’s Response.
Stationary sets
Definition
Suppose λ is an uncountable regular cardinal.
1. A set C ? λ is closed and unbounded if C is cofinal in λ
and C contains all of its limit points below λ:
I For all limit ordinals η < λ, if C ∩ η is cofinal in η then η ∈ C.
Stationary sets
Definition
Suppose λ is an uncountable regular cardinal.
1. A set C ? λ is closed and unbounded if C is cofinal in λ
and C contains all of its limit points below λ:
I For all limit ordinals η < λ, if C ∩ η is cofinal in η then η ∈ C.
2. A set S ? λ is stationary if S ∩ C 6= ? for all closed
unbounded sets C ? λ.
Stationary sets
Definition
Suppose λ is an uncountable regular cardinal.
1. A set C ? λ is closed and unbounded if C is cofinal in λ
and C contains all of its limit points below λ:
I For all limit ordinals η < λ, if C ∩ η is cofinal in η then η ∈ C.
2. A set S ? λ is stationary if S ∩ C 6= ? for all closed
unbounded sets C ? λ.
Example:
I Let S ? ω2 be the set all ordinals α such that cof(α) = ω.
I S is a stationary subset of ω2,
I ω2\S is a stationary subset of ω2.
The Solovay Splitting Theorem
Theorem (Solovay)
Suppose that λ is an uncountable regular cardinal and that S ? λ
is stationary.
I Then there is a partition
hSα : α < λi
of S into λ-many pairwise disjoint stationary subsets of λ.
The Solovay Splitting Theorem
Theorem (Solovay)
Suppose that λ is an uncountable regular cardinal and that S ? λ
is stationary.
I Then there is a partition
hSα : α < λi
of S into λ-many pairwise disjoint stationary subsets of λ.
But suppose S ∈ HOD.
I Can one require
Sα ∈ HOD
for all α < λ?
The Solovay Splitting Theorem
Theorem (Solovay)
Suppose that λ is an uncountable regular cardinal and that S ? λ
is stationary.
I Then there is a partition
hSα : α < λi
of S into λ-many pairwise disjoint stationary subsets of λ.
But suppose S ∈ HOD.
I Can one require
Sα ∈ HOD
for all α < λ?
I Or just find a partition of S into 2 stationary sets, each in
HOD?
Lemma
Suppose that λ is an uncountable regular cardinal and that:
I S ? λ is stationary.
I S ∈ HOD.
I κ < λ and (2κ
)
HOD ≥ λ.
Lemma
Suppose that λ is an uncountable regular cardinal and that:
I S ? λ is stationary.
I S ∈ HOD.
I κ < λ and (2κ
)
HOD ≥ λ.
Then there is a partition
hSα : α < κi
of S into κ-many pairwise disjoint stationary subsets of λ such that
hSα : α < κi ∈ HOD.
Lemma
Suppose that λ is an uncountable regular cardinal and that:
I S ? λ is stationary.
I S ∈ HOD.
I κ < λ and (2κ
)
HOD ≥ λ.
Then there is a partition
hSα : α < κi
of S into κ-many pairwise disjoint stationary subsets of λ such that
hSα : α < κi ∈ HOD.
But what if:
I S = {α < λ cof(α) = ω} and (2κ
)
HOD < λ?
Definition
Let λ be an uncountable regular cardinal and let
S = {α < λ cof(α) = ω}.
Then λ is ω-strongly measurable in HOD if there exists κ < λ
such that:
1. (2κ
)
HOD < λ,
2. there is no partition hSα | α < κi of S into stationary sets
such that
Sα ∈ HOD
for all α < λ.
Definition
Let λ be an uncountable regular cardinal and let
S = {α < λ cof(α) = ω}.
Then λ is ω-strongly measurable in HOD if there exists κ < λ
such that:
1. (2κ
)
HOD < λ,
2. there is no partition hSα | α < κi of S into stationary sets
such that
Sα ∈ HOD
for all α < λ.
Lemma
Assume λ is ω-strongly measurable in HOD. Then
HOD |= λ is a measurable cardinal.
Extendible cardinals
Lemma
Suppose that
π : Vα+1 → Vπ(α)+1
is an elementary embedding and π is not the identity.
I Then there exists an ordinal η that π(η) 6= η.
I CRT(π) denotes the least η such that π(η) 6= η.
Extendible cardinals
Lemma
Suppose that
π : Vα+1 → Vπ(α)+1
is an elementary embedding and π is not the identity.
I Then there exists an ordinal η that π(η) 6= η.
I CRT(π) denotes the least η such that π(η) 6= η.
Definition (Reinhardt)
Suppose that δ is a cardinal.
I Then δ is an extendible cardinal if for each λ > δ there
exists an elementary embedding
π : Vλ+1 → Vπ(λ)+1
such that CRT(π) = δ and π(δ) > λ.
Extendible cardinals and a dichotomy theorem
Theorem (HOD Dichotomy Theorem (weak version))
Suppose that δ is an extendible cardinal. Then one of the following
holds.
(1) No regular cardinal κ ≥ δ is ω-strongly measurable in HOD.
Further, suppose γ is a singular cardinal and γ > δ.
I Then γ is singular cardinal in HOD and γ
+ = (γ
+)
HOD.
Extendible cardinals and a dichotomy theorem
Theorem (HOD Dichotomy Theorem (weak version))
Suppose that δ is an extendible cardinal. Then one of the following
holds.
(1) No regular cardinal κ ≥ δ is ω-strongly measurable in HOD.
Further, suppose γ is a singular cardinal and γ > δ.
I Then γ is singular cardinal in HOD and γ
+ = (γ
+)
HOD.
(2) Every regular cardinal κ ≥ δ is ω-strongly measurable in
HOD.
Extendible cardinals and a dichotomy theorem
Theorem (HOD Dichotomy Theorem (weak version))
Suppose that δ is an extendible cardinal. Then one of the following
holds.
(1) No regular cardinal κ ≥ δ is ω-strongly measurable in HOD.
Further, suppose γ is a singular cardinal and γ > δ.
I Then γ is singular cardinal in HOD and γ
+ = (γ
+)
HOD.
(2) Every regular cardinal κ ≥ δ is ω-strongly measurable in
HOD.
I If there is an extendible cardinal then HOD must be either
close to V or HOD must be far from V.
I This is just like the Jensen Dichotomy Theorem but with
HOD in place of L.
Supercompactness
Definition
Suppose that κ is an uncountable regular cardinal and that κ < λ.
1. Pκ(λ) = {σ ? λ |σ| < κ}.
2. Suppose that U ? P (Pκ(λ)) is an ultrafilter.
I U is fine if for each α < λ,
{σ ∈ Pκ(λ) α ∈ σ} ∈ U.
Supercompactness
Definition
Suppose that κ is an uncountable regular cardinal and that κ < λ.
1. Pκ(λ) = {σ ? λ |σ| < κ}.
2. Suppose that U ? P (Pκ(λ)) is an ultrafilter.
I U is fine if for each α < λ,
{σ ∈ Pκ(λ) α ∈ σ} ∈ U.
I U is normal if for each function
f : Pκ(λ) → λ
such that
{σ ∈ Pκ(λ) f (σ) ∈ σ} ∈ U,
there exists α < λ such that
{σ ∈ Pκ(λ) f (σ) = α} ∈ U.
Supercompact cardinals
Definition (Solovay, Reinhardt)
Suppose that κ is an uncountable regular cardinal.
I Then κ is a supercompact cardinal if for each λ > κ there
exists an ultrafilter U on Pκ(λ) such that:
I U is κ-complete, normal, fine ultrafilter.
Supercompact cardinals
Definition (Solovay, Reinhardt)
Suppose that κ is an uncountable regular cardinal.
I Then κ is a supercompact cardinal if for each λ > κ there
exists an ultrafilter U on Pκ(λ) such that:
I U is κ-complete, normal, fine ultrafilter.
Lemma (Magidor)
Suppose that δ is strongly inaccessible. Then the following are
equivalent.
(1) δ is supercompact.
(2) For all λ > δ there exist δ <ˉ λ < δ ˉ and an elementary
embedding
π : Vλˉ+1 → Vλ+1
such that CRT(π) = δˉ and such that π(δˉ) = δ.
Supercompact cardinals and a dichotomy theorem
Theorem
Suppose that δ is an supercompact cardinal, κ > δ is a regular
cardinal, and that κ is ω-strongly measurable in HOD.
I Then every regular cardinal λ > 2
κ
is ω-strongly measurable
in HOD.
Supercompact cardinals and a dichotomy theorem
Theorem
Suppose that δ is an supercompact cardinal, κ > δ is a regular
cardinal, and that κ is ω-strongly measurable in HOD.
I Then every regular cardinal λ > 2
κ
is ω-strongly measurable
in HOD.
I Assuming δ is an extendible cardinal then one obtains a much
stronger conclusion.
Supercompact cardinals and the Singular Cardinals
Hypothesis
Theorem (Solovay)
Suppose that δ is a supercompact cardinal and that γ > δ is a
singular strong limit cardinal.
I Then 2
γ = γ
+.
Supercompact cardinals and the Singular Cardinals
Hypothesis
Theorem (Solovay)
Suppose that δ is a supercompact cardinal and that γ > δ is a
singular strong limit cardinal.
I Then 2
γ = γ
+.
Theorem (Silver)
Suppose that δ is a supercompact cardinal. Then there is a generic
extension V[G] of V such that in V[G]:
I δ is a supercompact cardinal.
I 2
δ > δ+.
Supercompact cardinals and the Singular Cardinals
Hypothesis
Theorem (Solovay)
Suppose that δ is a supercompact cardinal and that γ > δ is a
singular strong limit cardinal.
I Then 2
γ = γ
+.
Theorem (Silver)
Suppose that δ is a supercompact cardinal. Then there is a generic
extension V[G] of V such that in V[G]:
I δ is a supercompact cardinal.
I 2
δ > δ+.
I Solovay’s Theorem is the strongest possible theorem on
supercompact cardinals and the Generalized Continuum
Hypothesis.
The δ-cover and δ-approximation properties
Definition (Hamkins)
Suppose N is an inner model and that δ is an uncountable regular
cardinal of V.
1. N has the δ-cover property if for all σ ? N, if |σ| < δ then
there exists τ ? N such that:
I σ ? τ ,
I τ ∈ N,
I |τ | < δ.
The δ-cover and δ-approximation properties
Definition (Hamkins)
Suppose N is an inner model and that δ is an uncountable regular
cardinal of V.
1. N has the δ-cover property if for all σ ? N, if |σ| < δ then
there exists τ ? N such that:
I σ ? τ ,
I τ ∈ N,
I |τ | < δ.
2. N has the δ-approximation property if for all sets X ? N,
the following are equivalent.
I X ∈ N.
I For all σ ∈ N if |σ| < δ then σ ∩ X ∈ N.
The δ-cover and δ-approximation properties
Definition (Hamkins)
Suppose N is an inner model and that δ is an uncountable regular
cardinal of V.
1. N has the δ-cover property if for all σ ? N, if |σ| < δ then
there exists τ ? N such that:
I σ ? τ ,
I τ ∈ N,
I |τ | < δ.
2. N has the δ-approximation property if for all sets X ? N,
the following are equivalent.
I X ∈ N.
I For all σ ∈ N if |σ| < δ then σ ∩ X ∈ N.
For each (infinite) cardinal γ:
I H(γ) denotes the union of all transitive sets M such that
|M| < γ.
The Hamkins Uniqueness Theorem
Theorem (Hamkins)
Suppose N0 and N1 both have the δ-approximation property and
the δ-cover property. Suppose
I N0 ∩ H(δ
+) = N1 ∩ H(δ
+).
Then:
I N0 = N1.
The Hamkins Uniqueness Theorem
Theorem (Hamkins)
Suppose N0 and N1 both have the δ-approximation property and
the δ-cover property. Suppose
I N0 ∩ H(δ
+) = N1 ∩ H(δ
+).
Then:
I N0 = N1.
Corollary
Suppose N has the δ-approximation property and the δ-cover
property. Let A = N ∩ H(δ
+).
I Then N ∩ H(γ) is (uniformly) definable in H(γ) from A,
I for all strong limit cardinals γ > δ+.
I N is a Σ2-definable class from parameters.
Inner models with the δ-approximation property and the
δ-cover property are close to V
Theorem
Suppose N is an inner model with the δ-approximation property
and the δ-cover property.
I Suppose γ > δ and γ is a singular cardinal.
Then:
I γ is a singular cardinal in N.
I γ
+ = (γ
+)
N.
Set Theoretic Geology
Definition (Hamkins)
An inner model N is a ground of V if
I N |= ZFC.
I There is a partial order P ∈ N and an N-generic filter G ? P
such that V = N[G].
I G is allowed to be trivial in which case N = V.
Set Theoretic Geology
Definition (Hamkins)
An inner model N is a ground of V if
I N |= ZFC.
I There is a partial order P ∈ N and an N-generic filter G ? P
such that V = N[G].
I G is allowed to be trivial in which case N = V.
Lemma (Hamkins)
Suppose N is a ground of V. Then for all sufficiently large regular
cardinals δ:
I N has the δ-approximation property.
I N has the δ-cover property.
I Simply take δ be any regular cardinal of N such that |P|
N < δ.
Corollary
The grounds of V are Σ2-definable classes from parameters.
Corollary
The grounds of V are Σ2-definable classes from parameters.
I By the Hamkins Uniqueness Theorem.
Corollary
The grounds of V are Σ2-definable classes from parameters.
I By the Hamkins Uniqueness Theorem.
Set Theoretic Geology (Hamkins)
What is the possible structure of the grounds of V?
I This is part of the first order theory of V.
Corollary
The grounds of V are Σ2-definable classes from parameters.
I By the Hamkins Uniqueness Theorem.
Set Theoretic Geology (Hamkins)
What is the possible structure of the grounds of V?
I This is part of the first order theory of V.
I Suppose N ? M ? V, N is a ground of V, and M |= ZFC.
I Then M is a ground of V and N is a ground of M.
Corollary
The grounds of V are Σ2-definable classes from parameters.
I By the Hamkins Uniqueness Theorem.
Set Theoretic Geology (Hamkins)
What is the possible structure of the grounds of V?
I This is part of the first order theory of V.
I Suppose N ? M ? V, N is a ground of V, and M |= ZFC.
I Then M is a ground of V and N is a ground of M.
Definition (Hamkins)
The mantle of V is the intersection of all the grounds of V.
Corollary
The grounds of V are Σ2-definable classes from parameters.
I By the Hamkins Uniqueness Theorem.
Set Theoretic Geology (Hamkins)
What is the possible structure of the grounds of V?
I This is part of the first order theory of V.
I Suppose N ? M ? V, N is a ground of V, and M |= ZFC.
I Then M is a ground of V and N is a ground of M.
Definition (Hamkins)
The mantle of V is the intersection of all the grounds of V.
Let M be the mantle of V.
I (Hamkins) If M is a ground of V then M has no nontrivial
grounds.
Corollary
The grounds of V are Σ2-definable classes from parameters.
I By the Hamkins Uniqueness Theorem.
Set Theoretic Geology (Hamkins)
What is the possible structure of the grounds of V?
I This is part of the first order theory of V.
I Suppose N ? M ? V, N is a ground of V, and M |= ZFC.
I Then M is a ground of V and N is a ground of M.
Definition (Hamkins)
The mantle of V is the intersection of all the grounds of V.
Let M be the mantle of V.
I (Hamkins) If M is a ground of V then M has no nontrivial
grounds.
I (Hamkins) M |= ZF but must M |= ZFC?
The Directed Grounds Problem
Question (Hamkins)
Are the grounds of V downward set-directed under inclusion?
The Directed Grounds Problem
Question (Hamkins)
Are the grounds of V downward set-directed under inclusion?
Claim
Suppose that grounds of V are downwards set-directed. Then the
following are equivalent.
1. The mantle of V is a ground of V.
2. There are only set-many grounds of V.
3. This is a minimum ground of V.
The Directed Grounds Problem
Question (Hamkins)
Are the grounds of V downward set-directed under inclusion?
Claim
Suppose that grounds of V are downwards set-directed. Then the
following are equivalent.
1. The mantle of V is a ground of V.
2. There are only set-many grounds of V.
3. This is a minimum ground of V.
Claim
Suppose that grounds of V are downwards set-directed and let M
be the mantle of V. Then
M |= ZFC.
Bukovsky’s Theorem and Usuba’s Solution
Theorem (Bukovsky)
Suppose that κ is a regular cardinal and N ? V is an inner model.
Then the following are equivalent.
1. For each θ ∈ Ord and for each function F : θ → N there
exists a function
H : θ → Pκ(N)
such that H ∈ N and such that F(α) ∈ H(α) for all α < θ.
Bukovsky’s Theorem and Usuba’s Solution
Theorem (Bukovsky)
Suppose that κ is a regular cardinal and N ? V is an inner model.
Then the following are equivalent.
1. For each θ ∈ Ord and for each function F : θ → N there
exists a function
H : θ → Pκ(N)
such that H ∈ N and such that F(α) ∈ H(α) for all α < θ.
2. V is a κ-cc generic extension of N.
Bukovsky’s Theorem and Usuba’s Solution
Theorem (Bukovsky)
Suppose that κ is a regular cardinal and N ? V is an inner model.
Then the following are equivalent.
1. For each θ ∈ Ord and for each function F : θ → N there
exists a function
H : θ → Pκ(N)
such that H ∈ N and such that F(α) ∈ H(α) for all α < θ.
2. V is a κ-cc generic extension of N.
Theorem (Usuba)
The grounds of V are downward set-directed under inclusion.
Bukovsky’s Theorem and Usuba’s Solution
Theorem (Bukovsky)
Suppose that κ is a regular cardinal and N ? V is an inner model.
Then the following are equivalent.
1. For each θ ∈ Ord and for each function F : θ → N there
exists a function
H : θ → Pκ(N)
such that H ∈ N and such that F(α) ∈ H(α) for all α < θ.
2. V is a κ-cc generic extension of N.
Theorem (Usuba)
The grounds of V are downward set-directed under inclusion.
Corollary (Usuba)
Let M be the mantle of V.
I Then M |= The Axiom of Choice.
Usuba’s Mantle Theorem
Theorem (Usuba)
Suppose that there is an extendible cardinal. Let M be the mantle
of V.
I Then M is a ground of V.
Usuba’s Mantle Theorem
Theorem (Usuba)
Suppose that there is an extendible cardinal. Let M be the mantle
of V.
I Then M is a ground of V.
Corollary
Suppose that there is an extendible cardinal. Let M be the mantle
of V and suppose that M ? HOD.
I Then HOD is a ground of V.
Usuba’s Mantle Theorem
Theorem (Usuba)
Suppose that there is an extendible cardinal. Let M be the mantle
of V.
I Then M is a ground of V.
Corollary
Suppose that there is an extendible cardinal. Let M be the mantle
of V and suppose that M ? HOD.
I Then HOD is a ground of V.
I In this case, the far option in the HOD Dichotomy Theorem
cannot hold.
A natural conjecture
Assuming sufficient large cardinals exist, then provably the far
option in the HOD Dichotomy Theorem cannot hold.
The HOD Hypothesis
Definition (The HOD Hypothesis)
There exists a proper class of regular cardinals λ which are not
ω-strongly measurable in HOD.
The HOD Hypothesis
Definition (The HOD Hypothesis)
There exists a proper class of regular cardinals λ which are not
ω-strongly measurable in HOD.
I It is not known if there can exist 4 regular cardinals which are
ω-strongly measurable in HOD.
The HOD Hypothesis
Definition (The HOD Hypothesis)
There exists a proper class of regular cardinals λ which are not
ω-strongly measurable in HOD.
I It is not known if there can exist 4 regular cardinals which are
ω-strongly measurable in HOD.
I It is not known if there can exist 2 regular cardinals above 2?0
where are ω-strongly measurable in HOD.
The HOD Hypothesis
Definition (The HOD Hypothesis)
There exists a proper class of regular cardinals λ which are not
ω-strongly measurable in HOD.
I It is not known if there can exist 4 regular cardinals which are
ω-strongly measurable in HOD.
I It is not known if there can exist 2 regular cardinals above 2?0
where are ω-strongly measurable in HOD.
I Suppose γ is a singular cardinal of uncountable cofinality.
I It is not known if γ
+ can ever be ω-strongly measurable in
HOD.
The HOD Hypothesis
Definition (The HOD Hypothesis)
There exists a proper class of regular cardinals λ which are not
ω-strongly measurable in HOD.
I It is not known if there can exist 4 regular cardinals which are
ω-strongly measurable in HOD.
I It is not known if there can exist 2 regular cardinals above 2?0
where are ω-strongly measurable in HOD.
I Suppose γ is a singular cardinal of uncountable cofinality.
I It is not known if γ
+ can ever be ω-strongly measurable in
HOD.
Conjecture
Suppose γ > 2
?0 and that γ
+ is ω-strongly measurable in HOD.
I Then γ
++ is not ω-strongly measurable in HOD.
The HOD Conjecture
Definition (HOD Conjecture)
The theory
ZFC + “There is a supercompact cardinal”
proves the HOD Hypothesis.
The HOD Conjecture
Definition (HOD Conjecture)
The theory
ZFC + “There is a supercompact cardinal”
proves the HOD Hypothesis.
I Assume the HOD Conjecture and that there is an extendible
cardinal. Then:
I The far option in the HOD Dichotomy Theorem is vacuous:
I HOD must be close to V.
The HOD Conjecture
Definition (HOD Conjecture)
The theory
ZFC + “There is a supercompact cardinal”
proves the HOD Hypothesis.
I Assume the HOD Conjecture and that there is an extendible
cardinal. Then:
I The far option in the HOD Dichotomy Theorem is vacuous:
I HOD must be close to V.
I The HOD Conjecture is a number theoretic statement.
The Weak HOD Conjecture and the Ultimate-L
Conjecture
Definition (Weak HOD Conjecture)
The theory
ZFC + “There is a extendible cardinal”
proves the HOD Hypothesis.
The Weak HOD Conjecture and the Ultimate-L
Conjecture
Definition (Weak HOD Conjecture)
The theory
ZFC + “There is a extendible cardinal”
proves the HOD Hypothesis.
Ultimate-L Conjecture (weak version)
(ZFC) Suppose that δ is an extendible cardinal. Then (provably)
there is an inner model N such that:
1. N has the δ-approximation property and the δ-cover property.
2. N |= “V = Ultimate-L”.
The Weak HOD Conjecture and the Ultimate-L
Conjecture
Definition (Weak HOD Conjecture)
The theory
ZFC + “There is a extendible cardinal”
proves the HOD Hypothesis.
Ultimate-L Conjecture (weak version)
(ZFC) Suppose that δ is an extendible cardinal. Then (provably)
there is an inner model N such that:
1. N has the δ-approximation property and the δ-cover property.
2. N |= “V = Ultimate-L”.
Theorem
The Ultimate-L Conjecture implies the Weak HOD Conjecture.
An equivalence
Theorem
Suppose there is a proper class of extendible cardinals. Then
following are equivalent.
(1) The HOD Hypothesis holds.
(2) For some δ, there is an inner model N with the
δ-approximation property and the δ-cover property such that
N |= “The HOD Hypothesis”.
Weak extender models and universality
Definition
Suppose N is an inner model.
I Then N is a weak extender model of δ is supercompact if
for every γ > δ there exists a normal fine δ-complete
ultrafilter U on Pδ(γ) such thta:
I N ∩ Pδ(γ) ∈ U,
I U ∩ N ∈ N.
Weak extender models and universality
Definition
Suppose N is an inner model.
I Then N is a weak extender model of δ is supercompact if
for every γ > δ there exists a normal fine δ-complete
ultrafilter U on Pδ(γ) such thta:
I N ∩ Pδ(γ) ∈ U,
I U ∩ N ∈ N.
Universality Theorem (weak version)
Suppose N is a weak extender model of δ is supercompact and
that U is a δ-complete ultrafilter on λ for some λ ≥ δ.
I Then U ∩ N ∈ N.
The HOD Dichotomy (full version)
Theorem (HOD Dichotomy Theorem)
Suppose that δ is an extendible cardinal. Then one of the following
holds.
(1) No regular cardinal κ ≥ δ is ω-strongly measurable in HOD.
Further:
I HOD is a weak extender model of δ is supercompact.
The HOD Dichotomy (full version)
Theorem (HOD Dichotomy Theorem)
Suppose that δ is an extendible cardinal. Then one of the following
holds.
(1) No regular cardinal κ ≥ δ is ω-strongly measurable in HOD.
Further:
I HOD is a weak extender model of δ is supercompact.
(2) Every regular cardinal κ ≥ δ is ω-strongly measurable in HOD.
Further:
I HOD is not a weak extender of λ is supercompact, for any λ.
I There is no weak extender model N of λ is supercompact such
that N ? HOD, for any λ.
A unconditional corollary
Theorem
Suppose that δ is an extendible cardinal, κ ≥ δ, and that κ is a
measurable cardinal.
I Then κ is a measurable cardinal in HOD.
A unconditional corollary
Theorem
Suppose that δ is an extendible cardinal, κ ≥ δ, and that κ is a
measurable cardinal.
I Then κ is a measurable cardinal in HOD.
Two cases by appealing to the HOD Dichotomy Theorem:
I Case 1: HOD is close to V. Then HOD is a weak extender
model of δ is supercompact.
I Apply the Universality Theorem.
A unconditional corollary
Theorem
Suppose that δ is an extendible cardinal, κ ≥ δ, and that κ is a
measurable cardinal.
I Then κ is a measurable cardinal in HOD.
Two cases by appealing to the HOD Dichotomy Theorem:
I Case 1: HOD is close to V. Then HOD is a weak extender
model of δ is supercompact.
I Apply the Universality Theorem.
I Case 2: HOD is far from V. Then every regular cardinal
κ ≥ δ is a measurable cardinal in HOD;
I since κ is ω-strongly measurable in HOD.
Weak extender models, approximation, and cover
Theorem
Suppose N is a weak extender model of δ is supercompact.
I Then N has the δ-approximation property and the δ-cover
property.
Weak extender models, approximation, and cover
Theorem
Suppose N is a weak extender model of δ is supercompact.
I Then N has the δ-approximation property and the δ-cover
property.
Suppose N is a weak extender model of δ is supercompact. Thus:
I N is uniquely specified by N ∩ H(δ
+).
I N is Σ2-definable from N ∩ H(δ
+).
I The theory of weak extender models is part of the theory of V.
Weak extender models, approximation, and cover
Theorem
Suppose N is a weak extender model of δ is supercompact.
I Then N has the δ-approximation property and the δ-cover
property.
Suppose N is a weak extender model of δ is supercompact. Thus:
I N is uniquely specified by N ∩ H(δ
+).
I N is Σ2-definable from N ∩ H(δ
+).
I The theory of weak extender models is part of the theory of V.
Theorem
Suppose there is an extendible cardinal and that N is an inner
model. Then the following are equivalent.
I N has the δ-approximation property and the δ-cover property,
for some δ.
I N is a weak extender model of δ is supercompact, for some δ.
The δ-genericity property and strong universality
Definition
Suppose that N is an inner model and δ is an uncountable regular
cardinal.
I Then N has the δ-genericity property if for all σ ? δ, if
|σ| < δ then σ is N-generic for some partial P ∈ N such that
|P| < δ.
The δ-genericity property and strong universality
Definition
Suppose that N is an inner model and δ is an uncountable regular
cardinal.
I Then N has the δ-genericity property if for all σ ? δ, if
|σ| < δ then σ is N-generic for some partial P ∈ N such that
|P| < δ.
Suppose that δ is strongly inaccessible.
I Then HOD has the δ-genericity property.
The δ-genericity property and strong universality
Definition
Suppose that N is an inner model and δ is an uncountable regular
cardinal.
I Then N has the δ-genericity property if for all σ ? δ, if
|σ| < δ then σ is N-generic for some partial P ∈ N such that
|P| < δ.
Suppose that δ is strongly inaccessible.
I Then HOD has the δ-genericity property.
Theorem
Suppose there is an extendible cardinal and that
I N has the δ-approximation property, the δ-cover property, and
the δ-genericity property.
Suppose that the Axiom I0 holds at λ, for some λ > δ.
I Then in N, the Axiom I0 holds at λ, for some λ > δ.
A new family of inner models with the approximation and
cover properties
Theorem
Suppose N is a weak extender model of δ is supercompact and
that N has the δ-genericity property.
Suppose U ∈ Vδ is a countably complete ultrafilter and that
NU = Ult0(N,U).
Then:
I NU has the δ-cover property.
I NU has the δ-approximation property.
A new family of inner models with the approximation and
cover properties
Theorem
Suppose N is a weak extender model of δ is supercompact and
that N has the δ-genericity property.
Suppose U ∈ Vδ is a countably complete ultrafilter and that
NU = Ult0(N,U).
Then:
I NU has the δ-cover property.
I NU has the δ-approximation property.
I Assume δ is a strong cardinal and that N has the
δ-approximation property and the δ-cover property.
A new family of inner models with the approximation and
cover properties
Theorem
Suppose N is a weak extender model of δ is supercompact and
that N has the δ-genericity property.
Suppose U ∈ Vδ is a countably complete ultrafilter and that
NU = Ult0(N,U).
Then:
I NU has the δ-cover property.
I NU has the δ-approximation property.
I Assume δ is a strong cardinal and that N has the
δ-approximation property and the δ-cover property.
I NU has the δ-cover property.
A new family of inner models with the approximation and
cover properties
Theorem
Suppose N is a weak extender model of δ is supercompact and
that N has the δ-genericity property.
Suppose U ∈ Vδ is a countably complete ultrafilter and that
NU = Ult0(N,U).
Then:
I NU has the δ-cover property.
I NU has the δ-approximation property.
I Assume δ is a strong cardinal and that N has the
δ-approximation property and the δ-cover property.
I NU has the δ-cover property.
I NU can fail to have the δ-approximation property:
I Even if N = V.
Too close to be useful?
I Are weak extender models of supercompactness simply too
close to V to be of any use in the search for generalizations of
L?
Too close to be useful?
I Are weak extender models of supercompactness simply too
close to V to be of any use in the search for generalizations of
L?
Theorem (Kunen)
There is no nontrivial elementary embedding
π : Vλ+2 → Vλ+2.
Too close to be useful?
I Are weak extender models of supercompactness simply too
close to V to be of any use in the search for generalizations of
L?
Theorem (Kunen)
There is no nontrivial elementary embedding
π : Vλ+2 → Vλ+2.
Theorem
Suppose that N is a weak extender model of δ is supercompact
and λ > δ.
I Then there is no nontrivial elementary embedding
π : N ∩ Vλ+2 → N ∩ Vλ+2
such that CRT(π) ≥ δ.
Perhaps not
I Weak extender models of supercompactness can be
nontrivially far from V in one key sense.
Perhaps not
I Weak extender models of supercompactness can be
nontrivially far from V in one key sense.
Theorem (Kunen)
The following are equivalent.
1. L is far from V (as in the Jensen Dichotomy Theorem).
2. There is a nontrivial elementary embedding j : L → L.
Perhaps not
I Weak extender models of supercompactness can be
nontrivially far from V in one key sense.
Theorem (Kunen)
The following are equivalent.
1. L is far from V (as in the Jensen Dichotomy Theorem).
2. There is a nontrivial elementary embedding j : L → L.
Theorem
Suppose that δ is a supercompact cardinal.
I Then there exists a weak extender model N of δ is
supercompact such that
I N
ω ? N.
I There is a nontrivial elementary embedding j : N → N.
The Ultimate-L Conjecture
Ultimate-L Conjecture
(ZFC) Suppose that δ is an extendible cardinal. Then (provably)
there is a inner model N such that:
1. N is a weak extender model of δ is supercompact.
2. N has the δ-genericity property.
3. N |= “V = Ultimate-L”.
Applications of the HOD Conjecture in ZF
Theorem (ZF)
Assume the HOD Conjecture and that there is a proper class of
extendible cardinals.
I Suppose δ is an extendible cardinal.
Applications of the HOD Conjecture in ZF
Theorem (ZF)
Assume the HOD Conjecture and that there is a proper class of
extendible cardinals.
I Suppose δ is an extendible cardinal.
Then for every regular cardinal λ ≥ δ:
I λ
+ is a regular cardinal.
Applications of the HOD Conjecture in ZF
Theorem (ZF)
Assume the HOD Conjecture and that there is a proper class of
extendible cardinals.
I Suppose δ is an extendible cardinal.
Then for every regular cardinal λ ≥ δ:
I λ
+ is a regular cardinal.
I The Solovay Splitting Theorem holds at λ.
Applications of the HOD Conjecture in ZF
Theorem (ZF)
Assume the HOD Conjecture and that there is a proper class of
extendible cardinals.
I Suppose δ is an extendible cardinal.
Then for every regular cardinal λ ≥ δ:
I λ
+ is a regular cardinal.
I The Solovay Splitting Theorem holds at λ.
I Assuming the HOD Conjecture:
I Large cardinal axioms are trying to prove the Axiom of Choice.
Berkeley cardinals
Definition
A cardinal δ is a Berkeley cardinal if:
I For all α < δ and for all transitive sets M with δ ? M, there
exists a nontrivial elementary embedding
j : M → M
such that α < CRT(j) < δ.
Berkeley cardinals
Definition
A cardinal δ is a Berkeley cardinal if:
I For all α < δ and for all transitive sets M with δ ? M, there
exists a nontrivial elementary embedding
j : M → M
such that α < CRT(j) < δ.
I Assuming the Axiom of Choice, there are no Berkeley
cardinals by Kunen’s Theorem:
I Just let M = Vδ+2.
Berkeley cardinals
Definition
A cardinal δ is a Berkeley cardinal if:
I For all α < δ and for all transitive sets M with δ ? M, there
exists a nontrivial elementary embedding
j : M → M
such that α < CRT(j) < δ.
I Assuming the Axiom of Choice, there are no Berkeley
cardinals by Kunen’s Theorem:
I Just let M = Vδ+2.
Theorem (ZF)
Assume the HOD Conjecture. Then:
I There are no Berkeley cardinals.
Summary
There is a progression of theorems from large cardinal hypotheses
that suggest:
I Some version of V = L is true.
Further:
I The theorems become much stronger as the large cardinal
hypothesis is increased.
Summary
There is a progression of theorems from large cardinal hypotheses
that suggest:
I Some version of V = L is true.
Further:
I The theorems become much stronger as the large cardinal
hypothesis is increased.
Large cardinals amplify structure.
I They measure V and force the structure of V into
discrete options.
Summary
There is a progression of theorems from large cardinal hypotheses
that suggest:
I Some version of V = L is true.
Further:
I The theorems become much stronger as the large cardinal
hypothesis is increased.
Large cardinals amplify structure.
I They measure V and force the structure of V into
discrete options.
Perhaps this is all evidence that V = Ultimate-L.
