Recent Advances in Core Model Theory
Recent Advances in Core Model Theory
Open problems (under construction)

Determine the consistency strength of
"every real has a sharp + u_{2}
= ω_{2}" where
u_{2}
is the second uniform indiscernible.

Steel and Welch showed that a
strong cardinal is a lower bound.
It is not known how to use the hypothesis
u_{2} = ω_{2}
to build models with more large cardinals than this
even if granted that K exists.

An upper bound is the existence of δ < μ
where δ is a Woodin cardinal
and μ is a measurable cardinal.

A related problem is to determine the
consistency strength of
"every real has a sharp
+ every subset of ω_{1} in
L(R) is constructible from a real".
 References:
 Steel, J. R., and Welch., P. D.,
Σ^{1}_{3}
absoluteness
and the second uniform indiscernible,
Israel J. Math. 104 (1998) 157190
 Woodin, W. H.,
The axiom of determinacy, forcing axioms
and the nonstationary ideal,
DeGruyter series in logic and its applications,
vol. 1, 1999

Assume PD.
Then there is a largest countable Π^{1}_{3}
set of reals called C_{3} . It also follows from PD that
for each n < ω,
there is a minimal proper class model with n
Woodin cardinals called M_{n} .
Is it true that C_{3}
is the set of
reals that are
Δ^{1}_{3} equivalent to a mastercode of
M_{2}?

C_{2} is the set of reals in
M_{0} = L .

C_{1} is the set of reals that are
Δ^{1}_{1} equivalent to a
mastercode of M_{0} = L .

C_{2n} is the set of reals in M_{2n} .

The general question is how this pattern continues
at the odd levels.

References:

Kechris, A.,
The theory of countable analytic sets,
Trans. Amer. Math. Soc. 202 (1975) 259297

Steel, J. R.,
Projectively wellordered inner models,
Ann. Pure Appl. Logic 74 (1995) 77104

Working in ZFC, how large can Θ^{L(R)} be?

Clearly ω_{1}
< Θ^{L(R)} .

It is consistent that
ω_{2} < Θ^{L(R)} .
For example, if
u_{2} = ω_{2} .

But what about
ω_{3} < Θ^{L(R)}?

(Variant)
Assume there are arbitrarily large Woodin cardinals.
Is it possible that there is a universally Baire wellordering
of type ω_{3}?

Assume AD^{+} and that
there is no countable mouse with a superstrong cardinal.
Let x be an ordinal definable real.
Does there exist a countable mouse M with x ε M ?

Woodin obtained a positive answer under the
stronger assumption that
and that there is no countable mouse
for the AD_{R} hypothesis.

Assume that the
M_{1}^{#} operation is total on sets.
Suppose that M_{2} does not exist.
Then K exists and is closed under the
M_{1}^{#} operation.
Therefore K is Σ^{1}_{3} correct.
Is K is Σ^{1}_{4} correct?

The following is a related theorem of Steel.
Assume that the
sharp operation is total on sets.
Suppose that M_{1} does not exist.
Then K exists and is closed under the
sharp operation.
Suppose that there is a measurable cardinal.
Then K is Σ^{1}_{3} correct.

It is conjectured that a measurable cardinal is not
needed for Steel's
Σ^{1}_{3} correctness theorem.
 Steel's correctness
theorem extends earlier results
of Jensen, Magidor and Mitchell.

Reference:
 Steel, J. R.,
The core model iterability problem,
Springer, 1996

Assume that
the M_{1}^{#} operation is total.
Let N be the least
M_{1}^{#} closed model.
Suppose that there is a Π^{1}_{3}
singleton not in N.
Let {x} be the least such singleton.
Does N^{#}
exist and is there a
Δ^{1}_{3} isomorphism
between
x and N^{#}?

This would give Σ^{1}_{4}
correctness for K in the case that K does not go beyond N.

The second clause in the above conclusion is
an instance of problem 2.
 (Two part question.)

Let M be a countable transitive set.
Suppose that there exists α
and an elementary embedding
of M into V_{α} .
Is M (ω_{1}+ 1)iterable?

Let T be a
countable iteration tree of limit length on V
with extenders
E_{ξ} and models
M_{ξ}.
Assume that
E_{ξ} is
countably closed in M_{ξ}
for all ξ < lh(T).
Does T have a cofinal wellfounded branch?

This is an instance of CBH.

Countably closed means that
M_{ξ}
satisfies
"the ultrapower by
E_{ξ} is closed under ω sequences".

Woodin showed that CBH is false.

Extender models satisfy UBH.

Let L[E] be an extender model such that every countable
premouse that embeds into a level of L[E] is
(ω_{1}+1)iterable.
In terms of large cardinal axioms, characterize those
L[E] successor cardinals λ with the property that
every stationary subset of λ∩cof(ω)
reflects.

The iterability hypothesis is enough to conclude
that all the standard condensation lemmas apply to L[E].

A variant of the question asks about
λ∩cof(<κ)

Schimmerling has written a report
(PDF)
with results on this question.

What is the consistency strength of "λ is a singular
cardinal and weak square fails at λ"?

An upper bound is the existence of a cardinal κ
that is κ^{+ω}strongly compact.

An upper bound for the failure of square at
aleph_{ω} is the existence of
a measurable subcompact cardinal.
This points to a possible difference between
square and weak square.

Assume K exists.
Let j be an elementary embedding from V to a transitive class M.
Let i be the restriction of j to K.
Then i is an elementary embedding from V to j(K) = K^{M}.
Does i arise from an iteration of K?

Schindler proves instances of this
in his paper, Iterates of the core model.
 (Two part problem)

Rate the consistency strength of
the following statement.
Let I be a simply definable
σideal.
(E.g., the ideal of countable sets,
null sets, meager sets, etc.)
Then the statement
"Every Σ^{1}_{2}
(projective) Ipositive set has a Borel
Ipositive subset"
holds in every generic extension.

Assume 0^{#} does not exist.
Is it possible to add a real x by forcing
such that R^{L[x]} is Ipositive?

Assume that

V=W[r] for some real r,

V and W have the same cofinalities,

CH holds in W, and

the continuum is aleph_{2} in V.
Prove there is an inner model
with aleph_{2} many measurable cardinals.

Shelah showed that under this hypothesis,
there is an inner model with a measurable cardinal.

Investigate the ZFC model HOD^{V[G]}
where G is Vgeneric over Coll(ω, < OR).
In particular, does
CH hold in this model?

Assume 0Pistol does not exist.
Suppose κ is Mahlo
and Diamond_{κ} (Sing) fails.

Must κ be a measurable cardinal in K?

In addition, suppose that GCH holds below κ.
Is there an inner model with a strong cardinal?

Can GCH hold?

Starting with a model with
a measurable cardinal κ
with o(κ)=κ^{++}
+ ε,
Woodin produced a model
with a Mahlo cardinal κ
for which Diamond_{κ} fails.

Zeman showed that if κ is a Mahlo cardinal
and Diamond_{κ}(Sing) fails,
then for all λ < κ there
exists δ < κ such that
o^{K}(δ) > λ.

(Two part question.)

Assume there is no proper class
inner model with a Woodin cardinal.
Must there exist a set iterable
extender model with the weak covering property?

Assume ZFC + NS_{ω1}
is ω_{2} saturated.
Is there an inner model with a Woodin cardinal?

Let M be the minimal fully iterable
extender model with a Woodin cardinal κ
that is a limit of Woodin cardinals. Let D be the
derived model of M below κ. Is Θ^{D}
regular in D?

Determine the consistency strength of
incompatible models of AD^{+},
by which we mean that there are
A and B such that
L(A, R) and L(B,R) satisfy
AD^{+}
but L(A, B, R) does not satisfy AD.

Neeman and Woodin showed that a Woodin
limit of Woodin cardinals is an upper bound.

Woodin showed that AD_{R} + DC
is a lower bound.

Let Θ = Θ^{L(R)} and
δ be the least Woodin cardinal of
M_{ω}.

Is HOD^{L(R)}∩V_{Θ}
a normal iterate of
M_{ω}∩V_{δ}?
(Neeman has evidence towards a negative answer.)

If not, is there a normal iterate Q of
HOD^{L(R)}
such that the iteration map fixes
Θ and
Q∩V_{Θ}
is a normal iterate of every countable
iterate of
M_{ω}∩V_{δ}?

Assume V = L(R) and AD holds.
Let Γ be a Π^{1}_{1}
like scaled pointclass. (I.e.,
closed under ∀^{R} and not selfdual.)
Let Δ be the corresponding selfdual pointclass.
Let δ be the supremum of the lengths of Δ
prewellorderings.
Is Γ closed under unions of length less than δ?

For Π^{1}_{3},
this is a result of Kechris and Martin.

For Π^{1}_{2n+5},
this is a result of Jackson.

Is there an inner model M of L[0^{#}], an Mdefinable
poset P ⊂ M and an Mgeneric filter G over P such that

0^{#} is not an element of M,

0^{#} is an element of M[G], and

(M[G], ε, G) is a model of ZFC?