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Pseudo-Prikry Sequences
Chris Lambie-Hanson
Department of Mathematics and Applied Mathematics
Virginia Commonwealth University
Prikry Forcing Online
14 December 2020
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I. A brief history
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The basic premise
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The basic premise
Prikry-type forcings have been uniquely effective tools for proving
consistency results about cardinal arithmetic and combinatorics at
singular cardinals and their successors.
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The basic premise
Prikry-type forcings have been uniquely effective tools for proving
consistency results about cardinal arithmetic and combinatorics at
singular cardinals and their successors.
In the last 30 years, some results have given some indication
about why this might be the case.
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The basic premise
Prikry-type forcings have been uniquely effective tools for proving
consistency results about cardinal arithmetic and combinatorics at
singular cardinals and their successors.
In the last 30 years, some results have given some indication
about why this might be the case.
These results all have the following flavor:
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The basic premise
Prikry-type forcings have been uniquely effective tools for proving
consistency results about cardinal arithmetic and combinatorics at
singular cardinals and their successors.
In the last 30 years, some results have given some indication
about why this might be the case.
These results all have the following flavor:
If V is an inner model of W and there is a regular cardinal
in V that is singular in W (and certain other cardinals are
preserved from V to W ), then there is an object in W
that resembles a generic object over V for some Prikry-
type forcing.
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The basic premise
Prikry-type forcings have been uniquely effective tools for proving
consistency results about cardinal arithmetic and combinatorics at
singular cardinals and their successors.
In the last 30 years, some results have given some indication
about why this might be the case.
These results all have the following flavor:
If V is an inner model of W and there is a regular cardinal
in V that is singular in W (and certain other cardinals are
preserved from V to W ), then there is an object in W
that resembles a generic object over V for some Prikry-
type forcing.
We call such objects pseudo-Prikry sequences.
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Diagonalization
A key property of generic objects for Prikry-type objects is that
they diagonalize some ground model ultrafilter (or sequence of
ultrafilters).
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Diagonalization
A key property of generic objects for Prikry-type objects is that
they diagonalize some ground model ultrafilter (or sequence of
ultrafilters). For example, if U is a normal ultrafilter over κ and
PU is the usual Prikry forcing defined by using U, then a sequence
of ordinals αn | n < ω generates a PU-generic object if and only
if, for all X ∈ U, we have αn ∈ X for all sufficiently large n < ω.
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Diagonalization
A key property of generic objects for Prikry-type objects is that
they diagonalize some ground model ultrafilter (or sequence of
ultrafilters). For example, if U is a normal ultrafilter over κ and
PU is the usual Prikry forcing defined by using U, then a sequence
of ordinals αn | n < ω generates a PU-generic object if and only
if, for all X ∈ U, we have αn ∈ X for all sufficiently large n < ω.
In an abstract setting in which we just have models V ⊆ W and a
regular cardinal κ in V has been singularized in W , we may not
have a normal ultrafilter over κ in V to be diagonalized in W .
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Diagonalization
A key property of generic objects for Prikry-type objects is that
they diagonalize some ground model ultrafilter (or sequence of
ultrafilters). For example, if U is a normal ultrafilter over κ and
PU is the usual Prikry forcing defined by using U, then a sequence
of ordinals αn | n < ω generates a PU-generic object if and only
if, for all X ∈ U, we have αn ∈ X for all sufficiently large n < ω.
In an abstract setting in which we just have models V ⊆ W and a
regular cardinal κ in V has been singularized in W , we may not
have a normal ultrafilter over κ in V to be diagonalized in W .
There is a natural normal filter over κ to take its place, though:
the club filter on κ.
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Diagonalization
A key property of generic objects for Prikry-type objects is that
they diagonalize some ground model ultrafilter (or sequence of
ultrafilters). For example, if U is a normal ultrafilter over κ and
PU is the usual Prikry forcing defined by using U, then a sequence
of ordinals αn | n < ω generates a PU-generic object if and only
if, for all X ∈ U, we have αn ∈ X for all sufficiently large n < ω.
In an abstract setting in which we just have models V ⊆ W and a
regular cardinal κ in V has been singularized in W , we may not
have a normal ultrafilter over κ in V to be diagonalized in W .
There is a natural normal filter over κ to take its place, though:
the club filter on κ.
So our pseudo-Prikry sequences will be sequences in W which
appropriately diagonalize certain club filters as defined in V .
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The earliest results
Theorem (Dˇ
zamonja-Shelah, ‘95, [2])
Suppose that:
1 V is an inner model of W ;
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The earliest results
Theorem (Dˇ
zamonja-Shelah, ‘95, [2])
Suppose that:
1 V is an inner model of W ;
2 κ is an inaccessible cardinal in V and a singular cardinal of
cofinality θ in W ;
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The earliest results
Theorem (Dˇ
zamonja-Shelah, ‘95, [2])
Suppose that:
1 V is an inner model of W ;
2 κ is an inaccessible cardinal in V and a singular cardinal of
cofinality θ in W ;
3 (κ+)W = (κ+)V ;
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The earliest results
Theorem (Dˇ
zamonja-Shelah, ‘95, [2])
Suppose that:
1 V is an inner model of W ;
2 κ is an inaccessible cardinal in V and a singular cardinal of
cofinality θ in W ;
3 (κ+)W = (κ+)V ;
4 Cα | α < κ+ ∈ V is a sequence of clubs in κ.
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The earliest results
Theorem (Dˇ
zamonja-Shelah, ‘95, [2])
Suppose that:
1 V is an inner model of W ;
2 κ is an inaccessible cardinal in V and a singular cardinal of
cofinality θ in W ;
3 (κ+)W = (κ+)V ;
4 Cα | α < κ+ ∈ V is a sequence of clubs in κ.
Then, in W , there is a sequence γi | i < θ of ordinals such that,
for all α < κ+ and all sufficiently large i < θ, γi ∈ Cα.
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The earliest results
Theorem (Dˇ
zamonja-Shelah, ‘95, [2])
Suppose that:
1 V is an inner model of W ;
2 κ is an inaccessible cardinal in V and a singular cardinal of
cofinality θ in W ;
3 (κ+)W = (κ+)V ;
4 Cα | α < κ+ ∈ V is a sequence of clubs in κ.
Then, in W , there is a sequence γi | i < θ of ordinals such that,
for all α < κ+ and all sufficiently large i < θ, γi ∈ Cα.
A similar theorem is proven by Gitik [3].
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Inevitable square sequences
Cummings and Schimmerling [1] proved that, if G is a generic
filter over V for Prikry forcing at a cardinal κ, then κ,ω holds in
V [G].
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Inevitable square sequences
Cummings and Schimmerling [1] proved that, if G is a generic
filter over V for Prikry forcing at a cardinal κ, then κ,ω holds in
V [G]. It turns out that a pseudo-Prikry sequence as in the Gitik
or Dˇ
zamonja-Shelah theorems is enough to reach the same
conclusion.
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Inevitable square sequences
Cummings and Schimmerling [1] proved that, if G is a generic
filter over V for Prikry forcing at a cardinal κ, then κ,ω holds in
V [G]. It turns out that a pseudo-Prikry sequence as in the Gitik
or Dˇ
zamonja-Shelah theorems is enough to reach the same
conclusion.
Theorem (Gitik, Dˇ
zamonja-Shelah, Cummings-Schimmerling)
Suppose that:
1 V is an inner model of W ;
2 κ is an inaccessible cardinal in V and a singular cardinal of
cofinality ℵ0 in W ;
3 (κ+)W = (κ+)V .
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Inevitable square sequences
Cummings and Schimmerling [1] proved that, if G is a generic
filter over V for Prikry forcing at a cardinal κ, then κ,ω holds in
V [G]. It turns out that a pseudo-Prikry sequence as in the Gitik
or Dˇ
zamonja-Shelah theorems is enough to reach the same
conclusion.
Theorem (Gitik, Dˇ
zamonja-Shelah, Cummings-Schimmerling)
Suppose that:
1 V is an inner model of W ;
2 κ is an inaccessible cardinal in V and a singular cardinal of
cofinality ℵ0 in W ;
3 (κ+)W = (κ+)V .
Then κ,ω holds in W .
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A two-cardinal generalization
Magidor and Sinapova generalize the D˘
zamonja-Shelah result in
two ways, including the following extension to clubs in Pκ(κ+n).
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A two-cardinal generalization
Magidor and Sinapova generalize the D˘
zamonja-Shelah result in
two ways, including the following extension to clubs in Pκ(κ+n).
Theorem (Magidor-Sinapova, ’17, [6])
Suppose that n < ω and:
1 V is an inner model of W ;
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A two-cardinal generalization
Magidor and Sinapova generalize the D˘
zamonja-Shelah result in
two ways, including the following extension to clubs in Pκ(κ+n).
Theorem (Magidor-Sinapova, ’17, [6])
Suppose that n < ω and:
1 V is an inner model of W ;
2 κ is a regular cardinal in V and, for all m ≤ n, (κ+m)V has
countable cofinality in W ;
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A two-cardinal generalization
Magidor and Sinapova generalize the D˘
zamonja-Shelah result in
two ways, including the following extension to clubs in Pκ(κ+n).
Theorem (Magidor-Sinapova, ’17, [6])
Suppose that n < ω and:
1 V is an inner model of W ;
2 κ is a regular cardinal in V and, for all m ≤ n, (κ+m)V has
countable cofinality in W ;
3 (κ+)W = (κ+n+1)V and (ω1)W < κ;
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A two-cardinal generalization
Magidor and Sinapova generalize the D˘
zamonja-Shelah result in
two ways, including the following extension to clubs in Pκ(κ+n).
Theorem (Magidor-Sinapova, ’17, [6])
Suppose that n < ω and:
1 V is an inner model of W ;
2 κ is a regular cardinal in V and, for all m ≤ n, (κ+m)V has
countable cofinality in W ;
3 (κ+)W = (κ+n+1)V and (ω1)W < κ;
4 Dα | α < κ+n+1 ∈ V is a sequence of clubs in Pκ(κ+n).
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A two-cardinal generalization
Magidor and Sinapova generalize the D˘
zamonja-Shelah result in
two ways, including the following extension to clubs in Pκ(κ+n).
Theorem (Magidor-Sinapova, ’17, [6])
Suppose that n < ω and:
1 V is an inner model of W ;
2 κ is a regular cardinal in V and, for all m ≤ n, (κ+m)V has
countable cofinality in W ;
3 (κ+)W = (κ+n+1)V and (ω1)W < κ;
4 Dα | α < κ+n+1 ∈ V is a sequence of clubs in Pκ(κ+n).
Then, in W , there is an increasing sequence xi | i < ω of
elements of (Pκ(κ+n))V such that, for all α < κ+n+1 and all
sufficiently large i < ω, xi ∈ Dα.
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A two-cardinal generalization
Magidor and Sinapova generalize the D˘
zamonja-Shelah result in
two ways, including the following extension to clubs in Pκ(κ+n).
Theorem (Magidor-Sinapova, ’17, [6])
Suppose that n < ω and:
1 V is an inner model of W ;
2 κ is a regular cardinal in V and, for all m ≤ n, (κ+m)V has
countable cofinality in W ;
3 (κ+)W = (κ+n+1)V and (ω1)W < κ;
4 Dα | α < κ+n+1 ∈ V is a sequence of clubs in Pκ(κ+n).
Then, in W , there is an increasing sequence xi | i < ω of
elements of (Pκ(κ+n))V such that, for all α < κ+n+1 and all
sufficiently large i < ω, xi ∈ Dα.
This can be seen as a ”pseudo-” version of a generic sequence for
supercompact Prikry forcing using a normal measure on Pκ(κ+n).
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A further extension
Gitik extends this result to the general setting of Pκ(µ), under
some additional cardinal arithmetic assumptions.
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A further extension
Gitik extends this result to the general setting of Pκ(µ), under
some additional cardinal arithmetic assumptions.
Theorem (Gitik, ’18, [4])
Suppose that
1 V is an inner model of W ;
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A further extension
Gitik extends this result to the general setting of Pκ(µ), under
some additional cardinal arithmetic assumptions.
Theorem (Gitik, ’18, [4])
Suppose that
1 V is an inner model of W ;
2 in V , κ < µ are regular cardinals and µ<µ = µ;
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A further extension
Gitik extends this result to the general setting of Pκ(µ), under
some additional cardinal arithmetic assumptions.
Theorem (Gitik, ’18, [4])
Suppose that
1 V is an inner model of W ;
2 in V , κ < µ are regular cardinals and µ<µ = µ;
3 in W , there is a sequence Qn | n < ω of elements of
(Pκ(µ))V such that n<ω
Qn = µ;
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A further extension
Gitik extends this result to the general setting of Pκ(µ), under
some additional cardinal arithmetic assumptions.
Theorem (Gitik, ’18, [4])
Suppose that
1 V is an inner model of W ;
2 in V , κ < µ are regular cardinals and µ<µ = µ;
3 in W , there is a sequence Qn | n < ω of elements of
(Pκ(µ))V such that n<ω
Qn = µ;
4 in W , µ ≥ (2ω)+ and (µ+)V is a cardinal;
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A further extension
Gitik extends this result to the general setting of Pκ(µ), under
some additional cardinal arithmetic assumptions.
Theorem (Gitik, ’18, [4])
Suppose that
1 V is an inner model of W ;
2 in V , κ < µ are regular cardinals and µ<µ = µ;
3 in W , there is a sequence Qn | n < ω of elements of
(Pκ(µ))V such that n<ω
Qn = µ;
4 in W , µ ≥ (2ω)+ and (µ+)V is a cardinal;
5 Dα | α < µ+ ∈ V is a sequence of clubs in Pκ(µ).
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A further extension
Gitik extends this result to the general setting of Pκ(µ), under
some additional cardinal arithmetic assumptions.
Theorem (Gitik, ’18, [4])
Suppose that
1 V is an inner model of W ;
2 in V , κ < µ are regular cardinals and µ<µ = µ;
3 in W , there is a sequence Qn | n < ω of elements of
(Pκ(µ))V such that n<ω
Qn = µ;
4 in W , µ ≥ (2ω)+ and (µ+)V is a cardinal;
5 Dα | α < µ+ ∈ V is a sequence of clubs in Pκ(µ).
Then, in W , there is an increasing sequence xi | i < ω of
elements of (Pκ(µ))V such that, for all α < µ+ and all
sufficiently large i < ω, xi ∈ Dα.
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II. PCF-theoretic background
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<∗-increasing sequences
Definition
Suppose that θ is a regular cardinal and f , g ∈ θOn. Then f <∗ g
if {i < θ | g(i) ≤ f (i)} is bounded in θ.
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<∗-increasing sequences
Definition
Suppose that θ is a regular cardinal and f , g ∈ θOn. Then f <∗ g
if {i < θ | g(i) ≤ f (i)} is bounded in θ. Expressions such as
f ≤∗ g and f =∗ g are defined in the obvious way.
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<∗-increasing sequences
Definition
Suppose that θ is a regular cardinal and f , g ∈ θOn. Then f <∗ g
if {i < θ | g(i) ≤ f (i)} is bounded in θ. Expressions such as
f ≤∗ g and f =∗ g are defined in the obvious way.
Definition (Exact upper bound)
Suppose that θ is a regular cardinal and f = fα | α < λ is a
<∗-increasing sequence of elements of θOn.
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<∗-increasing sequences
Definition
Suppose that θ is a regular cardinal and f , g ∈ θOn. Then f <∗ g
if {i < θ | g(i) ≤ f (i)} is bounded in θ. Expressions such as
f ≤∗ g and f =∗ g are defined in the obvious way.
Definition (Exact upper bound)
Suppose that θ is a regular cardinal and f = fα | α < λ is a
<∗-increasing sequence of elements of θOn. A function g ∈ θOn
is an exact upper bound (e.u.b.) for f if
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<∗-increasing sequences
Definition
Suppose that θ is a regular cardinal and f , g ∈ θOn. Then f <∗ g
if {i < θ | g(i) ≤ f (i)} is bounded in θ. Expressions such as
f ≤∗ g and f =∗ g are defined in the obvious way.
Definition (Exact upper bound)
Suppose that θ is a regular cardinal and f = fα | α < λ is a
<∗-increasing sequence of elements of θOn. A function g ∈ θOn
is an exact upper bound (e.u.b.) for f if
1 fα <∗ g for all α < λ;
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<∗-increasing sequences
Definition
Suppose that θ is a regular cardinal and f , g ∈ θOn. Then f <∗ g
if {i < θ | g(i) ≤ f (i)} is bounded in θ. Expressions such as
f ≤∗ g and f =∗ g are defined in the obvious way.
Definition (Exact upper bound)
Suppose that θ is a regular cardinal and f = fα | α < λ is a
<∗-increasing sequence of elements of θOn. A function g ∈ θOn
is an exact upper bound (e.u.b.) for f if
1 fα <∗ g for all α < λ;
2 for every function h ∈ θOn, if h <∗ g, then there is α < λ
such that h < fα.
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Scales
Definition
Suppose that µ is a singular cardinal and µi | i < cf(µ) is an
increasing sequence of regular cardinals, cofinal in µ.
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Scales
Definition
Suppose that µ is a singular cardinal and µi | i < cf(µ) is an
increasing sequence of regular cardinals, cofinal in µ. A sequence
f = fα | α < λ of functions in i
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Scales
Definition
Suppose that µ is a singular cardinal and µi | i < cf(µ) is an
increasing sequence of regular cardinals, cofinal in µ. A sequence
f = fα | α < λ of functions in i
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Scales
Definition
Suppose that µ is a singular cardinal and µi | i < cf(µ) is an
increasing sequence of regular cardinals, cofinal in µ. A sequence
f = fα | α < λ of functions in i
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Scales
Definition
Suppose that µ is a singular cardinal and µi | i < cf(µ) is an
increasing sequence of regular cardinals, cofinal in µ. A sequence
f = fα | α < λ of functions in i
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Club-increasing sequences
Definition
If θ is a regular cardinal and f = fα | α < λ is a sequence of
elements of θOn, we say that f is club-increasing if
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Club-increasing sequences
Definition
If θ is a regular cardinal and f = fα | α < λ is a sequence of
elements of θOn, we say that f is club-increasing if
1 f is <∗-increasing;
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Club-increasing sequences
Definition
If θ is a regular cardinal and f = fα | α < λ is a sequence of
elements of θOn, we say that f is club-increasing if
1 f is <∗-increasing;
2 for every limit ordinal γ < λ, there is a club D ⊆ γ and an
i < θ such that fα
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Club-increasing sequences
Definition
If θ is a regular cardinal and f = fα | α < λ is a sequence of
elements of θOn, we say that f is club-increasing if
1 f is <∗-increasing;
2 for every limit ordinal γ < λ, there is a club D ⊆ γ and an
i < θ such that fα
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Club-increasing sequences
Definition
If θ is a regular cardinal and f = fα | α < λ is a sequence of
elements of θOn, we say that f is club-increasing if
1 f is <∗-increasing;
2 for every limit ordinal γ < λ, there is a club D ⊆ γ and an
i < θ such that fα ν for all
i < θ.
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III. Diagonal pseudo-Prikry sequences
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Avoiding squares
Suppose one wants to find a model with a singular cardinal κ of
countable cofinality at which both SCH and κ,ω fail.
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Avoiding squares
Suppose one wants to find a model with a singular cardinal κ of
countable cofinality at which both SCH and κ,ω fail. The
simplest way to obtain a failure of SCH is to start with a
measurable cardinal κ at which GCH fails and do Prikry forcing.
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Avoiding squares
Suppose one wants to find a model with a singular cardinal κ of
countable cofinality at which both SCH and κ,ω fail. The
simplest way to obtain a failure of SCH is to start with a
measurable cardinal κ at which GCH fails and do Prikry forcing.
But we’ve seen that this necessarily forces κ,ω to hold.
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Avoiding squares
Suppose one wants to find a model with a singular cardinal κ of
countable cofinality at which both SCH and κ,ω fail. The
simplest way to obtain a failure of SCH is to start with a
measurable cardinal κ at which GCH fails and do Prikry forcing.
But we’ve seen that this necessarily forces κ,ω to hold. By a
generalization of this result due to Magidor and Sinapova, κ,ω
will hold in any extension W in which
• (κ+)W is the successor of a regular cardinal in V ;
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Avoiding squares
Suppose one wants to find a model with a singular cardinal κ of
countable cofinality at which both SCH and κ,ω fail. The
simplest way to obtain a failure of SCH is to start with a
measurable cardinal κ at which GCH fails and do Prikry forcing.
But we’ve seen that this necessarily forces κ,ω to hold. By a
generalization of this result due to Magidor and Sinapova, κ,ω
will hold in any extension W in which
• (κ+)W is the successor of a regular cardinal in V ;
• every V -regular cardinal in the interval [κ, (κ+)V ) has
countable cofinality in W .
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Avoiding squares
Suppose one wants to find a model with a singular cardinal κ of
countable cofinality at which both SCH and κ,ω fail. The
simplest way to obtain a failure of SCH is to start with a
measurable cardinal κ at which GCH fails and do Prikry forcing.
But we’ve seen that this necessarily forces κ,ω to hold. By a
generalization of this result due to Magidor and Sinapova, κ,ω
will hold in any extension W in which
• (κ+)W is the successor of a regular cardinal in V ;
• every V -regular cardinal in the interval [κ, (κ+)V ) has
countable cofinality in W .
This might lead us to look for Prikry-type extensions W in which
(κ+)W is the successor of a singular cardinal in V .
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Diagonal supercompact Prikry forcing
Solving this problem led Gitik and Sharon to introduce diagonal
supercompact Prikry forcing.
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Diagonal supercompact Prikry forcing
Solving this problem led Gitik and Sharon to introduce diagonal
supercompact Prikry forcing. Suppose that κ is a supercompact
cardinal, U∗ is a supercompactness measure on Pκ(κ+ω+1), and
Un is the projection of U∗ on Pκ(κ+n) for all n < ω.
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Diagonal supercompact Prikry forcing
Solving this problem led Gitik and Sharon to introduce diagonal
supercompact Prikry forcing. Suppose that κ is a supercompact
cardinal, U∗ is a supercompactness measure on Pκ(κ+ω+1), and
Un is the projection of U∗ on Pκ(κ+n) for all n < ω. The diagonal
supercompact Prikry forcing P associated to Un | n < ω
introduces an increasing sequence xn | n < ω such that
• xn ∈ Pκ(κ+n) for all n < ω;
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Diagonal supercompact Prikry forcing
Solving this problem led Gitik and Sharon to introduce diagonal
supercompact Prikry forcing. Suppose that κ is a supercompact
cardinal, U∗ is a supercompactness measure on Pκ(κ+ω+1), and
Un is the projection of U∗ on Pκ(κ+n) for all n < ω. The diagonal
supercompact Prikry forcing P associated to Un | n < ω
introduces an increasing sequence xn | n < ω such that
• xn ∈ Pκ(κ+n) for all n < ω;
• for every sequence Xn | n < ω ∈ V such that Xn ∈ Un for
all n < ω, we have xn ∈ Xn for all sufficiently large n < ω.
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Diagonal supercompact Prikry forcing
Solving this problem led Gitik and Sharon to introduce diagonal
supercompact Prikry forcing. Suppose that κ is a supercompact
cardinal, U∗ is a supercompactness measure on Pκ(κ+ω+1), and
Un is the projection of U∗ on Pκ(κ+n) for all n < ω. The diagonal
supercompact Prikry forcing P associated to Un | n < ω
introduces an increasing sequence xn | n < ω such that
• xn ∈ Pκ(κ+n) for all n < ω;
• for every sequence Xn | n < ω ∈ V such that Xn ∈ Un for
all n < ω, we have xn ∈ Xn for all sufficiently large n < ω.
As a result, in V P, cf((κ+n)V ) = ω for all n < ω.
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Diagonal supercompact Prikry forcing
Solving this problem led Gitik and Sharon to introduce diagonal
supercompact Prikry forcing. Suppose that κ is a supercompact
cardinal, U∗ is a supercompactness measure on Pκ(κ+ω+1), and
Un is the projection of U∗ on Pκ(κ+n) for all n < ω. The diagonal
supercompact Prikry forcing P associated to Un | n < ω
introduces an increasing sequence xn | n < ω such that
• xn ∈ Pκ(κ+n) for all n < ω;
• for every sequence Xn | n < ω ∈ V such that Xn ∈ Un for
all n < ω, we have xn ∈ Xn for all sufficiently large n < ω.
As a result, in V P, cf((κ+n)V ) = ω for all n < ω. κ remains a
cardinal, and (κ+)V P = (κ+ω+1)V .
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Diagonal supercompact Prikry forcing
Solving this problem led Gitik and Sharon to introduce diagonal
supercompact Prikry forcing. Suppose that κ is a supercompact
cardinal, U∗ is a supercompactness measure on Pκ(κ+ω+1), and
Un is the projection of U∗ on Pκ(κ+n) for all n < ω. The diagonal
supercompact Prikry forcing P associated to Un | n < ω
introduces an increasing sequence xn | n < ω such that
• xn ∈ Pκ(κ+n) for all n < ω;
• for every sequence Xn | n < ω ∈ V such that Xn ∈ Un for
all n < ω, we have xn ∈ Xn for all sufficiently large n < ω.
As a result, in V P, cf((κ+n)V ) = ω for all n < ω. κ remains a
cardinal, and (κ+)V P = (κ+ω+1)V . Moreover, APκ (and hence
κ,ω) fails in V P.
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Diagonal supercompact Prikry forcing
Solving this problem led Gitik and Sharon to introduce diagonal
supercompact Prikry forcing. Suppose that κ is a supercompact
cardinal, U∗ is a supercompactness measure on Pκ(κ+ω+1), and
Un is the projection of U∗ on Pκ(κ+n) for all n < ω. The diagonal
supercompact Prikry forcing P associated to Un | n < ω
introduces an increasing sequence xn | n < ω such that
• xn ∈ Pκ(κ+n) for all n < ω;
• for every sequence Xn | n < ω ∈ V such that Xn ∈ Un for
all n < ω, we have xn ∈ Xn for all sufficiently large n < ω.
As a result, in V P, cf((κ+n)V ) = ω for all n < ω. κ remains a
cardinal, and (κ+)V P = (κ+ω+1)V . Moreover, APκ (and hence
κ,ω) fails in V P. If 2κ = κ+ω+2 in V , then SCH fails at κ in V P
as well.
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Meeting diagonal clubs
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Meeting diagonal clubs
Definition
Suppose that µ is a singular cardinal and µ = µi | i < cf(µ) is
an increasing sequence of regular cardinals cofinal in µ.
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Meeting diagonal clubs
Definition
Suppose that µ is a singular cardinal and µ = µi | i < cf(µ) is
an increasing sequence of regular cardinals cofinal in µ. A
diagonal club in µ is a sequence Di | i < cf(µ) such that Di is a
club in µi for all i < cf(µ).
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Meeting diagonal clubs
Definition
Suppose that µ is a singular cardinal and µ = µi | i < cf(µ) is
an increasing sequence of regular cardinals cofinal in µ. A
diagonal club in µ is a sequence Di | i < cf(µ) such that Di is a
club in µi for all i < cf(µ).
Theorem (LH, ’18, [5])
Suppose that V is an inner model of W and, in V , µ is a singular
cardinal, cf(µ) = θ, and µ = µi | i < θ is an increasing sequence of
regular cardinals cofinal in µ such that there is a scale of length µ+ in
µ.
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Meeting diagonal clubs
Definition
Suppose that µ is a singular cardinal and µ = µi | i < cf(µ) is
an increasing sequence of regular cardinals cofinal in µ. A
diagonal club in µ is a sequence Di | i < cf(µ) such that Di is a
club in µi for all i < cf(µ).
Theorem (LH, ’18, [5])
Suppose that V is an inner model of W and, in V , µ is a singular
cardinal, cf(µ) = θ, and µ = µi | i < θ is an increasing sequence of
regular cardinals cofinal in µ such that there is a scale of length µ+ in
µ. Suppose moreover that, in W , (µ+)V is the successor of a
singular cardinal ν of cofinality θ and cf(µi
) = θ for all i < θ.
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Meeting diagonal clubs
Definition
Suppose that µ is a singular cardinal and µ = µi | i < cf(µ) is
an increasing sequence of regular cardinals cofinal in µ. A
diagonal club in µ is a sequence Di | i < cf(µ) such that Di is a
club in µi for all i < cf(µ).
Theorem (LH, ’18, [5])
Suppose that V is an inner model of W and, in V , µ is a singular
cardinal, cf(µ) = θ, and µ = µi | i < θ is an increasing sequence of
regular cardinals cofinal in µ such that there is a scale of length µ+ in
µ. Suppose moreover that, in W , (µ+)V is the successor of a
singular cardinal ν of cofinality θ and cf(µi
) = θ for all i < θ.
Then, in W , there is a function g ∈ µ such that, for every diagonal
club in µ, Di | i < θ ∈ V , we have g(i) ∈ Di
for all sufficiently large
i < θ.
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Meeting diagonal clubs
Definition
Suppose that µ is a singular cardinal and µ = µi | i < cf(µ) is
an increasing sequence of regular cardinals cofinal in µ. A
diagonal club in µ is a sequence Di | i < cf(µ) such that Di is a
club in µi for all i < cf(µ).
Theorem (LH, ’18, [5])
Suppose that V is an inner model of W and, in V , µ is a singular
cardinal, cf(µ) = θ, and µ = µi | i < θ is an increasing sequence of
regular cardinals cofinal in µ such that there is a scale of length µ+ in
µ. Suppose moreover that, in W , (µ+)V is the successor of a
singular cardinal ν of cofinality θ and cf(µi
) = θ for all i < θ.
Then, in W , there is a function g ∈ µ such that, for every diagonal
club in µ, Di | i < θ ∈ V , we have g(i) ∈ Di
for all sufficiently large
i < θ. Also, we can require sup{cf(g(i)) | i < θ} = ν.
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Proof
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More proof
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Still more proof
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This proof isn’t over yet?
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A variant
Notice that the essential point in the previous proof was the fact
that the ground-model scale was bounded by a function in µ in
W .
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A variant
Notice that the essential point in the previous proof was the fact
that the ground-model scale was bounded by a function in µ in
W . This leads to the following variant, which doesn’t require any
singularizing of cardinals.
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A variant
Notice that the essential point in the previous proof was the fact
that the ground-model scale was bounded by a function in µ in
W . This leads to the following variant, which doesn’t require any
singularizing of cardinals.
Theorem (LH, ’18, [5])
Suppose that V is an inner model of W and, in both V and W , µ
is a singular cardinal of cofinality θ.
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A variant
Notice that the essential point in the previous proof was the fact
that the ground-model scale was bounded by a function in µ in
W . This leads to the following variant, which doesn’t require any
singularizing of cardinals.
Theorem (LH, ’18, [5])
Suppose that V is an inner model of W and, in both V and W , µ
is a singular cardinal of cofinality θ. Suppose also that, in V ,
µ = µi | i < θ is an increasing sequence of regular cardinals
cofinal in µ such that there is a scale of length µ+ in µ.
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A variant
Notice that the essential point in the previous proof was the fact
that the ground-model scale was bounded by a function in µ in
W . This leads to the following variant, which doesn’t require any
singularizing of cardinals.
Theorem (LH, ’18, [5])
Suppose that V is an inner model of W and, in both V and W , µ
is a singular cardinal of cofinality θ. Suppose also that, in V ,
µ = µi | i < θ is an increasing sequence of regular cardinals
cofinal in µ such that there is a scale of length µ+ in µ.
Suppose finally that (µ+)V = (µ+)W and ( µ)V is bounded in
(( µ)W , <∗).
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A variant
Notice that the essential point in the previous proof was the fact
that the ground-model scale was bounded by a function in µ in
W . This leads to the following variant, which doesn’t require any
singularizing of cardinals.
Theorem (LH, ’18, [5])
Suppose that V is an inner model of W and, in both V and W , µ
is a singular cardinal of cofinality θ. Suppose also that, in V ,
µ = µi | i < θ is an increasing sequence of regular cardinals
cofinal in µ such that there is a scale of length µ+ in µ.
Suppose finally that (µ+)V = (µ+)W and ( µ)V is bounded in
(( µ)W , <∗).
Then, in W , there is a function g ∈ µ such that, for every
diagonal club in µ, Di | i < θ ∈ V , we have g(i) ∈ Di for all
sufficiently large i < θ. Also, we can require that
sup{cf(g(i)) | i < θ} = µ.
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Generalized diagonal sequences
Theorem (LH, ‘18, [5])
Suppose that:
1 V is an inner model of W ;
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Generalized diagonal sequences
Theorem (LH, ‘18, [5])
Suppose that:
1 V is an inner model of W ;
2 in V , cf(µ) = θ < κ = cf(κ) < µ are cardinals, with µ strong
limit;
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Generalized diagonal sequences
Theorem (LH, ‘18, [5])
Suppose that:
1 V is an inner model of W ;
2 in V , cf(µ) = θ < κ = cf(κ) < µ are cardinals, with µ strong
limit;
3 in V , µ = µi | i < θ is an increasing sequence of regular
cardinals, cofinal in µ, with κ ≤ µ0;
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Generalized diagonal sequences
Theorem (LH, ‘18, [5])
Suppose that:
1 V is an inner model of W ;
2 in V , cf(µ) = θ < κ = cf(κ) < µ are cardinals, with µ strong
limit;
3 in V , µ = µi | i < θ is an increasing sequence of regular
cardinals, cofinal in µ, with κ ≤ µ0;
4 in W , there is a ⊆-increasing sequence xi | i < θ from
(Pκ(µ))V such that i<θ
xi = µ;
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Generalized diagonal sequences
Theorem (LH, ‘18, [5])
Suppose that:
1 V is an inner model of W ;
2 in V , cf(µ) = θ < κ = cf(κ) < µ are cardinals, with µ strong
limit;
3 in V , µ = µi | i < θ is an increasing sequence of regular
cardinals, cofinal in µ, with κ ≤ µ0;
4 in W , there is a ⊆-increasing sequence xi | i < θ from
(Pκ(µ))V such that i<θ
xi = µ;
5 in W , (µ+)V remains a cardinal and µ ≥ 2θ;
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Generalized diagonal sequences
Theorem (LH, ‘18, [5])
Suppose that:
1 V is an inner model of W ;
2 in V , cf(µ) = θ < κ = cf(κ) < µ are cardinals, with µ strong
limit;
3 in V , µ = µi | i < θ is an increasing sequence of regular
cardinals, cofinal in µ, with κ ≤ µ0;
4 in W , there is a ⊆-increasing sequence xi | i < θ from
(Pκ(µ))V such that i<θ
xi = µ;
5 in W , (µ+)V remains a cardinal and µ ≥ 2θ;
6 in V , D(α) | α < µ+ is a sequence of diagonal clubs in
Pκ(µ).
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Generalized diagonal sequences
Theorem (LH, ‘18, [5])
Suppose that:
1 V is an inner model of W ;
2 in V , cf(µ) = θ < κ = cf(κ) < µ are cardinals, with µ strong
limit;
3 in V , µ = µi | i < θ is an increasing sequence of regular
cardinals, cofinal in µ, with κ ≤ µ0;
4 in W , there is a ⊆-increasing sequence xi | i < θ from
(Pκ(µ))V such that i<θ
xi = µ;
5 in W , (µ+)V remains a cardinal and µ ≥ 2θ;
6 in V , D(α) | α < µ+ is a sequence of diagonal clubs in
Pκ(µ).
Then, in W , there is yi | i < θ such that, for all α < µ+ and all
sufficiently large i < θ, yi ∈ D(α)i .
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IV. Fat trees and pseudo-Prikry sequences
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A new proof
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More new proof
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Still more new proof
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This had better be the end of the new proof
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Fat trees
Definition
Suppose that κ is a regular, uncountable cardinal, n < ω, and, for all
m ≤ n, λm ≥ κ is a regular cardinal. Then
T ⊆
k≤n+1 m
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Fat trees
Definition
Suppose that κ is a regular, uncountable cardinal, n < ω, and, for all
m ≤ n, λm ≥ κ is a regular cardinal. Then
T ⊆
k≤n+1 m
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Fat trees
Definition
Suppose that κ is a regular, uncountable cardinal, n < ω, and, for all
m ≤ n, λm ≥ κ is a regular cardinal. Then
T ⊆
k≤n+1 m
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Fat trees
Definition
Suppose that κ is a regular, uncountable cardinal, n < ω, and, for all
m ≤ n, λm ≥ κ is a regular cardinal. Then
T ⊆
k≤n+1 m
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Outside guessing of fat trees
Theorem (LH, ’18, [5])
Suppose that:
1 V is an inner model of W ;
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Outside guessing of fat trees
Theorem (LH, ’18, [5])
Suppose that:
1 V is an inner model of W ;
2 in V , κ ≤ λ are cardinals, with κ regular;
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Outside guessing of fat trees
Theorem (LH, ’18, [5])
Suppose that:
1 V is an inner model of W ;
2 in V , κ ≤ λ are cardinals, with κ regular;
3 in W , θ < θ+2 < |κ|, θ is a regular cardinal, and there is a
⊆-increasing sequence xi | i < θ from (Pκ(λ))V such that
i<θ
xi = λ;
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Outside guessing of fat trees
Theorem (LH, ’18, [5])
Suppose that:
1 V is an inner model of W ;
2 in V , κ ≤ λ are cardinals, with κ regular;
3 in W , θ < θ+2 < |κ|, θ is a regular cardinal, and there is a
⊆-increasing sequence xi | i < θ from (Pκ(λ))V such that
i<θ
xi = λ;
4 (λ+)V remains a cardinal in W ;
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Outside guessing of fat trees
Theorem (LH, ’18, [5])
Suppose that:
1 V is an inner model of W ;
2 in V , κ ≤ λ are cardinals, with κ regular;
3 in W , θ < θ+2 < |κ|, θ is a regular cardinal, and there is a
⊆-increasing sequence xi | i < θ from (Pκ(λ))V such that
i<θ
xi = λ;
4 (λ+)V remains a cardinal in W ;
5 n < ω and, in V , λi | i ≤ n is a sequence of regular
cardinals from [κ, λ] and T(α) | α < λ+ is a sequence of
fat trees of type (κ, λ0, . . . , λn ).
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Outside guessing of fat trees
Theorem (LH, ’18, [5])
Suppose that:
1 V is an inner model of W ;
2 in V , κ ≤ λ are cardinals, with κ regular;
3 in W , θ < θ+2 < |κ|, θ is a regular cardinal, and there is a
⊆-increasing sequence xi | i < θ from (Pκ(λ))V such that
i<θ
xi = λ;
4 (λ+)V remains a cardinal in W ;
5 n < ω and, in V , λi | i ≤ n is a sequence of regular
cardinals from [κ, λ] and T(α) | α < λ+ is a sequence of
fat trees of type (κ, λ0, . . . , λn ).
Then, in W , there is a sequence σi | i < θ such that, for all
α < λ+ and all sufficiently large i < θ, σi is a maximal element of
T(α).
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Some remarks
The proof of the theorem on the previous slide is an elaboration
of the proof presented at the beginning of this section.
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Some remarks
The proof of the theorem on the previous slide is an elaboration
of the proof presented at the beginning of this section.
The theorem yields as special cases all of the results mentioned in
the first section of this talk (though requiring very slightly
stronger hypotheses in some cases).
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Some remarks
The proof of the theorem on the previous slide is an elaboration
of the proof presented at the beginning of this section.
The theorem yields as special cases all of the results mentioned in
the first section of this talk (though requiring very slightly
stronger hypotheses in some cases).
Questions remain about the extent to which the hypotheses of
these results necessarily hold in outer models in which cardinals
have been singularized.
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Some remarks
The proof of the theorem on the previous slide is an elaboration
of the proof presented at the beginning of this section.
The theorem yields as special cases all of the results mentioned in
the first section of this talk (though requiring very slightly
stronger hypotheses in some cases).
Questions remain about the extent to which the hypotheses of
these results necessarily hold in outer models in which cardinals
have been singularized.
Conjecture (Gitik)
Suppose that V is an inner model of W , κ is regular in V ,
(cf(κ))W = ω, (ℵ1)V = (ℵ1)W , and V and W agree about a final
segment of cardinals. Then there is an inner model V ⊆ V such
that W contains a sequence that is generic over V for Namba
forcing, stationary tower forcing, or a Prikry-type forcing.
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References
James Cummings and Ernest Schimmerling, Indexed squares,
Israel J. Math. 131 (2002), 61–99. MR 1942302
Mirna Dˇ
zamonja and Saharon Shelah, On squares, outside
guessing of clubs and I
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All artwork by Victor Vasarely.
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Thank you!