Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Preserving sharps by tree forcings
Fabiana Castiblanco
Joint work with Philipp Schlicht

Institut für mathematische Logik und Grundlagenforschung

Infinite Combinatorics and Forcing Theory Workshop
Kyoto, December 1st

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

BackgroundWe say that M is an Inner model if it is transitive, satisfies ZFC and contains
all the ordinals.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

BackgroundWe say that M is an Inner model if it is transitive, satisfies ZFC and contains
all the ordinals.
Let M, N be inner models. We say that j : M ! N is an elementaryembedding if for every formula '(x) and for all sets a,
V |= '[a] if and only if M |= '[j(a)]Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

BackgroundWe say that M is an Inner model if it is transitive, satisfies ZFC and contains
all the ordinals.
Let M, N be inner models. We say that j : M ! N is an elementaryembedding if for every formula '(x) and for all sets a,
V |= '[a] if and only if M |= '[j(a)]Large cardinals are strong axioms of infinity. Their existence, in each case, isequivalent to the existence of some especial elementary embedding.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Motivation: Large cardinals and forcing

We say that  is a measurable cardinal if there is an elementary embeddingj : V ! M into a transitive model M such that crit(j) = .

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Motivation: Large cardinals and forcing

We say that  is a measurable cardinal if there is an elementary embeddingj : V ! M into a transitive model M such that crit(j) = .
Theorem (Levy-Solovay)
If P is a forcing notion of size <  and  is a measurable cardinal in V then remains to be measurable in V P .

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Motivation: Large cardinals and forcing

We say that  is a measurable cardinal if there is an elementary embeddingj : V ! M into a transitive model M such that crit(j) = .
Theorem (Levy-Solovay)
If P is a forcing notion of size <  and  is a measurable cardinal in V then remains to be measurable in V P .
There are variants to this theorem stating that compact, supercompact, strong,huge, Woodin, among others, are preserved a�er doing “small” forcing.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Motivation: Large cardinals and forcing

We say that  is a measurable cardinal if there is an elementary embeddingj : V ! M into a transitive model M such that crit(j) = .
Theorem (Levy-Solovay)
If P is a forcing notion of size <  and  is a measurable cardinal in V then remains to be measurable in V P .
There are variants to this theorem stating that compact, supercompact, strong,huge, Woodin, among others, are preserved a�er doing “small” forcing.
�estion: Do small forcing notions preserve consequences of large cardinals in thesame way?

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Consequences of large cardinals and forcing
If there exists a non-trivial elementary embedding j : V ! M observe thatj L : L ! L is also non-trivial and elementary.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Consequences of large cardinals and forcing
If there exists a non-trivial elementary embedding j : V ! M observe thatj L : L ! L is also non-trivial and elementary.
Definition
We say that 0# exists if there exists a non-trivial elementary j : L ! L

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Consequences of large cardinals and forcing
If there exists a non-trivial elementary embedding j : V ! M observe thatj L : L ! L is also non-trivial and elementary.
Definition
We say that 0# exists if there exists a non-trivial elementary j : L ! L
More generally, if x is a set of ordinals we say that x # exists i� there is a non-trivial
elementary embedding j : L[x] ! L[x] that does no move ordinals up to sup(x).

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Consequences of large cardinals and forcing
If there exists a non-trivial elementary embedding j : V ! M observe thatj L : L ! L is also non-trivial and elementary.
Definition
We say that 0# exists if there exists a non-trivial elementary j : L ! L
More generally, if x is a set of ordinals we say that x # exists i� there is a non-trivial
elementary embedding j : L[x] ! L[x] that does no move ordinals up to sup(x).
Theorem (folklore)
The property “For every set of ordinals x, x # exists” is preserved by any forcing.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Sharps for reals and forcing
�estion: Does any forcing notion preserves sharps for reals?

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Sharps for reals and forcing
�estion: Does any forcing notion preserves sharps for reals?
Theorem (R. David)
It is consistent that every real has a sharp and there is a ⌃13 -c.c.c. forcing notion
such that in the generic extension holds V = L[x] for some real x.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Sharps for reals and forcing
�estion: Does any forcing notion preserves sharps for reals?
Theorem (R. David)
It is consistent that every real has a sharp and there is a ⌃13 -c.c.c. forcing notion
such that in the generic extension holds V = L[x] for some real x.
However, if we impose some conditions over the forcing notion some positive
results hold:Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Sharps for reals and forcing
�estion: Does any forcing notion preserves sharps for reals?
Theorem (R. David)
It is consistent that every real has a sharp and there is a ⌃13 -c.c.c. forcing notion
such that in the generic extension holds V = L[x] for some real x.
However, if we impose some conditions over the forcing notion some positive
results hold:
Theorem (Schlicht)
Suppose that P ✓ R is a provably ⌃12 -definable c.c.c. forcing notion. Then, P
preserves the property “every real has a sharp”.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Our results�estion: Does any provably ⌃12 proper forcing notion preserve sharps for reals?

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Our results�estion: Does any provably ⌃12 proper forcing notion preserve sharps for reals?
We consider tree forcing notions such as Sacks (M), Silver (U), Mathias (M), Laver(L) and Miller forcing (ML).

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Our results�estion: Does any provably ⌃12 proper forcing notion preserve sharps for reals?
We consider tree forcing notions such as Sacks (M), Silver (U), Mathias (M), Laver(L) and Miller forcing (ML). In each of this forcing notions, the conditions are
perfect subtrees of <! 2 or <! ! ordered by inclusion. All these forcings satisfy
Axiom A and therefore, are proper.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Our results�estion: Does any provably ⌃12 proper forcing notion preserve sharps for reals?
We consider tree forcing notions such as Sacks (M), Silver (U), Mathias (M), Laver(L) and Miller forcing (ML). In each of this forcing notions, the conditions are
perfect subtrees of <! 2 or <! ! ordered by inclusion. All these forcings satisfy
Axiom A and therefore, are proper.
Theorem (C.-Schlicht)
Suppose P 2 {S, M, U, L, ML}. Then, P preserves the property “x # exists forevery real x”. Furthermore, if Mn# (x) exists for all x 2 R, n 2 !, then the same
holds in the generic extension. Equivalently, projective determinacy is preserved by
P.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Trees

Let (T , <) be a tree and for every t 2 T , let SuccT (t) be the set of all immediate
successors of t 2 T .
Let Split(T ) = {t 2 T : |SuccT (t)| > 1} and let Stem(T ) be the first element of
Split(T ).

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Trees

Let (T , <) be a tree and for every t 2 T , let SuccT (t) be the set of all immediate
successors of t 2 T .
Let Split(T ) = {t 2 T : |SuccT (t)| > 1} and let Stem(T ) be the first element of
Split(T ).
We say that T is perfect if for every s 2 T there exists t 2 Split(T ) such that s < t.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Arboreal forcing notions

Definition
A partial order P is arboreal if its conditions are perfect trees on ! or 2 ordered byinclusion. A partial order P is strongly arboreal if it is arboreal and for all T 2 P, if
t 2 T , Tt = {s 2 T : either s ✓ t or t ✓ s} 2 P.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Arboreal forcing notions

Definition
A partial order P is arboreal if its conditions are perfect trees on ! or 2 ordered byinclusion. A partial order P is strongly arboreal if it is arboreal and for all T 2 P, if
t 2 T , Tt = {s 2 T : either s ✓ t or t ✓ s} 2 P.
If P is strongly arboreal, we can code
Sgeneric objects by realsTin the standard way:
if G is P-generic over V , then xG = {Stem(T ) : T 2 G} = {[T ] : T 2 G} is areal and G = {T 2 P : xG 2 [T ]}Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Sacks forcing

S = {T : T is a perfect tree on 2}

For S, T 2 S we stipulate S  T if and only if S ✓ T .

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Sacks forcing

S = {T : T is a perfect tree on 2}

For S, T 2 S we stipulate S  T if and only if S ✓ T .
Definition
Suppose that S 2 S. We define:

A(S)S = {t ✓ S : t is a finite subtree of S isomorphic to some n 2 }ordered by end-extension, i.e. t  s if and only if t ◆ s and t |s| = s.
Given S 2 S, let ⇡S : Split(S) !<! 2 be the natural order isomorphism.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Lemma
Suppose that G is A(S)S -generic over V . Then:S
TG = G is a perfect subtree of S.
S
For every x 2 [TG ], ⇡S (x) := n<! ⇡S (x n ) is Cohen-generic over V .

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Lemma
Suppose that G is A(S)S -generic over V . Then:S
TG = G is a perfect subtree of S.
S
For every x 2 [TG ], ⇡S (x) := n<! ⇡S (x n ) is Cohen-generic over V .
Lemma (C.-S.)
Suppose that ẋ is a name for a S-generic real. Let 2 H(!1 ) be an S-name for areal and let S 2 S. Suppose that M is an inner model with , S 2 M and|}(A(S)S )M | < !1 . Then, there is some T  S such thatT

Fabiana Castiblanco
Preserving sharps by tree forcings

S ẋ is C-generic over M modulo ⇡S

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Lemma (C.-S.)
Suppose that V is closed under sharps for reals. Suppose that r 2 R. Then, forevery S-generic real x over V , there exists some real y 2 V such that x is C-genericover L[r, y].

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Lemma (C.-S.)
Suppose that V is closed under sharps for reals. Suppose that r 2 R. Then, forevery S-generic real x over V , there exists some real y 2 V such that x is C-genericover L[r, y].
Proof.
Suppose ẋ is a S-name for x. As (r, S)# exists, by the previous lemma applied tothe model L[r, S], the set
D = {T 2 S : for some S 2 S, T  S, T

S ẋ is C-generic over L[r, S]}

is dense in S.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Lemma (C.-S.)
Suppose that V is closed under sharps for reals. Suppose that r 2 R. Then, forevery S-generic real x over V , there exists some real y 2 V such that x is C-genericover L[r, y].
Proof.
Suppose ẋ is a S-name for x. As (r, S)# exists, by the previous lemma applied tothe model L[r, S], the set
D = {T 2 S : for some S 2 S, T  S, T

S ẋ is C-generic over L[r, S]}

is dense in S. If G is S-generic over V containing T 2 D, we can pick S
T

Fabiana Castiblanco
Preserving sharps by tree forcings

S ẋ

T withis C-generic over L[r, S] modulo ⇡S

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Lemma (C.-S.)
Suppose that V is closed under sharps for reals. Suppose that r 2 R. Then, forevery S-generic real x over V , there exists some real y 2 V such that x is C-genericover L[r, y].
Proof.
Suppose ẋ is a S-name for x. As (r, S)# exists, by the previous lemma applied tothe model L[r, S], the set
D = {T 2 S : for some S 2 S, T  S, T

S ẋ is C-generic over L[r, S]}

is dense in S. If G is S-generic over V containing T 2 D, we can pick S
TS ẋ

T withis C-generic over L[r, S] modulo ⇡S

Therefore, V [G] |= x is C-generic over L[r, S] modulo ⇡S .

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Theorem (C.-S.)
Suppose that 8y 2 R(y # exists). Suppose that x is S-generic over V . Then
V [x] |= 8y 2 R(y # exists)

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Theorem (C.-S.)
Suppose that 8y 2 R(y # exists). Suppose that x is S-generic over V . Then
V [x] |= 8y 2 R(y # exists)
Proof.
Let y 2 R \ V [x] and let

Fabiana Castiblanco
Preserving sharps by tree forcings

2 H(!1 ) a S-name such that

x

= y.

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Theorem (C.-S.)
Suppose that 8y 2 R(y # exists). Suppose that x is S-generic over V . Then
V [x] |= 8y 2 R(y # exists)
Proof.
Let y 2 R \ V [x] and let

2 H(!1 ) a S-name such that

x

= y.

By lemma above, there exists z 2 V \ R such that x is C-generic over L[ , z]
modulo ⇡z .

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Theorem (C.-S.)
Suppose that 8y 2 R(y # exists). Suppose that x is S-generic over V . Then
V [x] |= 8y 2 R(y # exists)
Proof.
Let y 2 R \ V [x] and let

2 H(!1 ) a S-name such that

x

= y.

By lemma above, there exists z 2 V \ R such that x is C-generic over L[ , z]
modulo ⇡z . Code the pair ( , z) 2 V by some real r. Then, we have in V anontrivial elementary embedding j : L[r] ! L[r].

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Theorem (C.-S.)
Suppose that 8y 2 R(y # exists). Suppose that x is S-generic over V . Then
V [x] |= 8y 2 R(y # exists)
Proof.
Let y 2 R \ V [x] and let

2 H(!1 ) a S-name such that

x

= y.

By lemma above, there exists z 2 V \ R such that x is C-generic over L[ , z]
modulo ⇡z . Code the pair ( , z) 2 V by some real r. Then, we have in V anontrivial elementary embedding j : L[r] ! L[r].
Li� this embedding to the Cohen extension:
j 0 : L[r][x] ! L[r][x]⌧ x ! (j(⌧ ))xFabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Theorem (C.-S.)
Suppose that 8y 2 R(y # exists). Suppose that x is S-generic over V . Then
V [x] |= 8y 2 R(y # exists)
Proof.
Let y 2 R \ V [x] and let

2 H(!1 ) a S-name such that

x

= y.

By lemma above, there exists z 2 V \ R such that x is C-generic over L[ , z]
modulo ⇡z . Code the pair ( , z) 2 V by some real r. Then, we have in V anontrivial elementary embedding j : L[r] ! L[r].
Li� this embedding to the Cohen extension:
j 0 : L[r][x] ! L[r][x]⌧ x ! (j(⌧ ))xNote that j 0 is elementary and non-trivial.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Theorem (C.-S.)
Suppose that 8y 2 R(y # exists). Suppose that x is S-generic over V . Then
V [x] |= 8y 2 R(y # exists)
Proof.
Let y 2 R \ V [x] and let

2 H(!1 ) a S-name such that

x

= y.

By lemma above, there exists z 2 V \ R such that x is C-generic over L[ , z]
modulo ⇡z . Code the pair ( , z) 2 V by some real r. Then, we have in V anontrivial elementary embedding j : L[r] ! L[r].
Li� this embedding to the Cohen extension:
j 0 : L[r][x] ! L[r][x]⌧ x ! (j(⌧ ))xNote that j 0 is elementary and non-trivial. Then j̄ := j 0 L[y] witness the existence ofy # in V [x].
Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Silver forcing
A Silver tree (or uniform tree) is a perfect tree T ✓ <! 2 such that for all s, t 2 T of
the same length we haves_ 0 2 T i� t _ 0 2 T and s_ 1 2 T i� t _ 1 2 T

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Silver forcing
A Silver tree (or uniform tree) is a perfect tree T ✓ <! 2 such that for all s, t 2 T of
the same length we haves_ 0 2 T i� t _ 0 2 T and s_ 1 2 T i� t _ 1 2 T
Let U be the collection of all Silver trees. For S, T 2 U we stipulate S  T if andonly if S ✓ T .

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Silver forcing
A Silver tree (or uniform tree) is a perfect tree T ✓ <! 2 such that for all s, t 2 T of
the same length we haves_ 0 2 T i� t _ 0 2 T and s_ 1 2 T i� t _ 1 2 T
Let U be the collection of all Silver trees. For S, T 2 U we stipulate S  T if andonly if S ✓ T .
Definition
Suppose that S ✓! 2 is an uniform tree.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Silver forcing
A Silver tree (or uniform tree) is a perfect tree T ✓ <! 2 such that for all s, t 2 T of
the same length we haves_ 0 2 T i� t _ 0 2 T and s_ 1 2 T i� t _ 1 2 T
Let U be the collection of all Silver trees. For S, T 2 U we stipulate S  T if andonly if S ✓ T .
Definition
Suppose that S ✓! 2 is an uniform tree. We define
A(U)S = {t ✓ S : t is a finite uniform tree of S}
If t 2 A(U)S , let ht(t) be the length of any branch through t and let
t n = {p 2 t : |p|  n}.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Silver forcing
A Silver tree (or uniform tree) is a perfect tree T ✓ <! 2 such that for all s, t 2 T of
the same length we haves_ 0 2 T i� t _ 0 2 T and s_ 1 2 T i� t _ 1 2 T
Let U be the collection of all Silver trees. For S, T 2 U we stipulate S  T if andonly if S ✓ T .
Definition
Suppose that S ✓! 2 is an uniform tree. We define
A(U)S = {t ✓ S : t is a finite uniform tree of S}
If t 2 A(U)S , let ht(t) be the length of any branch through t and let
t n = {p 2 t : |p|  n}. In A(U)S we stipulate s  t if and only if ht(s)and s ht(t) = t.

Fabiana Castiblanco
Preserving sharps by tree forcingsht(t)Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Proposition
Suppose that G is A(U)S generic over some model V . Then TG =subtree of S whose branches are C-generic over V modulo ⇡S .

Fabiana Castiblanco
Preserving sharps by tree forcings#
About Mn (x)Laver forcing

S

Applications

G is a Silver

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing#
About Mn (x)Laver forcing

Proposition
Suppose that G is A(U)S generic over some model V . Then TG =subtree of S whose branches are C-generic over V modulo ⇡S .

S

Applications

G is a Silver

Lemma (C.-S.)
Suppose that 8x 2 R(x # exists) and let 2 H(!1 ). Let ẏ a name for a Silver real.
For every S 2 U, there is some T  S such thatT

Fabiana Castiblanco
Preserving sharps by tree forcings

U ẏ is C-generic over L[, S] modulo ⇡S

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing#
About Mn (x)Laver forcing

Proposition
Suppose that G is A(U)S generic over some model V . Then TG =subtree of S whose branches are C-generic over V modulo ⇡S .

S

Applications

G is a Silver

Lemma (C.-S.)
Suppose that 8x 2 R(x # exists) and let 2 H(!1 ). Let ẏ a name for a Silver real.
For every S 2 U, there is some T  S such thatT

U ẏ is C-generic over L[, S] modulo ⇡S

By proceeding in the same way that in Sacks forcing case (li�ing the embedding to
the Cohen extension) we have that Silver forcing also preserves sharps for reals.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Mathias forcing
Let M = {(s, A) : s 2 [!]<! , A 2 [!]! , m«ax(s) < m«ın(A)}. In M, we stipulate thefollowing order:(s, A)  (t, B) () t ✓ s, A ✓ B and s r t ✓ B

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Mathias forcing
Let M = {(s, A) : s 2 [!]<! , A 2 [!]! , m«ax(s) < m«ın(A)}. In M, we stipulate thefollowing order:(s, A)  (t, B) () t ✓ s, A ✓ B and s r t ✓ B
Suppose that a ✓ n. Let

Fabiana Castiblanco
Preserving sharps by tree forcingsna denote the characteristic functionna : n ! 2.

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Mathias forcing
Let M = {(s, A) : s 2 [!]<! , A 2 [!]! , m«ax(s) < m«ın(A)}. In M, we stipulate thefollowing order:(s, A)  (t, B) () t ✓ s, A ✓ B and s r t ✓ B
Suppose that a ✓ n. Letna denote the characteristic functionna : n ! 2.

Proposition
For each (s, A) 2 M, letT(s,A) := { na k : s ✓ a ✓ s [ A, a 2 [!]<! , k  n 2 !}Let Mtree = {T(s,A) : (s, A) 2 M}. Then ⇡ : M ! Mtree defined as ⇡(s, A) = T(s,A)
is an isomorphism.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Lemma (C.-S.)
Suppose that Ġ is a name for a M-generic filter. Let p 2 M and suppose that M isan inner model with , p 2 M such that |}(M)M | < !1 . Then, there is some q  psuch thatq S Ġ is M-generic over M

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Lemma (C.-S.)
Suppose that Ġ is a name for a M-generic filter. Let p 2 M and suppose that M isan inner model with , p 2 M such that |}(M)M | < !1 . Then, there is some q  psuch thatq S Ġ is M-generic over M
Lemma (C.-S.)
Suppose that V is closed under sharps for reals and let r 2 R \ V . Then for eachM-generic real x over V there exists some y 2 R \ V such that x is M-generic overL[r, y].

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Lemma (C.-S.)
Suppose that Ġ is a name for a M-generic filter. Let p 2 M and suppose that M isan inner model with , p 2 M such that |}(M)M | < !1 . Then, there is some q  psuch thatq S Ġ is M-generic over M
Lemma (C.-S.)
Suppose that V is closed under sharps for reals and let r 2 R \ V . Then for eachM-generic real x over V there exists some y 2 R \ V such that x is M-generic overL[r, y].
Proceed as before...

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Laver forcing

A tree T ✓<! [!] is called a Laver tree if it has a stem s and 8t 2 T (t ✓ s or
|SuccT (t)| = !).

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Laver forcing

A tree T ✓<! [!] is called a Laver tree if it has a stem s and 8t 2 T (t ✓ s or
|SuccT (t)| = !). Laver forcing, denoted by L, is the set of all Laver trees ordered byinclusion.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Laver forcing

A tree T ✓<! [!] is called a Laver tree if it has a stem s and 8t 2 T (t ✓ s or
|SuccT (t)| = !). Laver forcing, denoted by L, is the set of all Laver trees ordered byinclusion.
We define an injection f : <! ! ! <! 2 by stipulating
f (t) = 0t(0) 10t(1) . . . 10t(n) 1,

Fabiana Castiblanco
Preserving sharps by tree forcingst 2 !!

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Laver forcing

A tree T ✓<! [!] is called a Laver tree if it has a stem s and 8t 2 T (t ✓ s or
|SuccT (t)| = !). Laver forcing, denoted by L, is the set of all Laver trees ordered byinclusion.
We define an injection f : <! ! ! <! 2 by stipulating
f (t) = 0t(0) 10t(1) . . . 10t(n) 1,

t 2 !!

Note that f (t) codes the values of t as blocks of 0 separated by 1.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Lemma
Let S 2 L, 2 H(!1 ) and consider f : <! ! ! <! 2 as above. Suppose that u ✓ !
is Mathias generic over L[ , S]. Then, T̄ = f 1 [T(?,u) ] is a Laver tree with stem ?.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing#
About Mn (x)Laver forcing

Applications

Lemma
Let S 2 L, 2 H(!1 ) and consider f : <! ! ! <! 2 as above. Suppose that u ✓ !
is Mathias generic over L[ , S]. Then, T̄ = f 1 [T(?,u) ] is a Laver tree with stem ?.
Lemma
Suppose that V is closed under sharps for reals. Let ẋ denote a name for a
L-generic real. Then for every S 2 L, 2 H(!1 ) there is some T  S such thatT

Fabiana Castiblanco
Preserving sharps by tree forcings

L ẋ is equivalent to a Mathias generic over L[, S]

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing#
About Mn (x)Laver forcing

Applications

Lemma
Let S 2 L, 2 H(!1 ) and consider f : <! ! ! <! 2 as above. Suppose that u ✓ !
is Mathias generic over L[ , S]. Then, T̄ = f 1 [T(?,u) ] is a Laver tree with stem ?.
Lemma
Suppose that V is closed under sharps for reals. Let ẋ denote a name for a
L-generic real. Then for every S 2 L, 2 H(!1 ) there is some T  S such thatT

L ẋ is equivalent to a Mathias generic over L[, S]

Mutatis mutandis...

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing#
About Mn (x)Laver forcing

Applications

Lemma
Let S 2 L, 2 H(!1 ) and consider f : <! ! ! <! 2 as above. Suppose that u ✓ !
is Mathias generic over L[ , S]. Then, T̄ = f 1 [T(?,u) ] is a Laver tree with stem ?.
Lemma
Suppose that V is closed under sharps for reals. Let ẋ denote a name for a
L-generic real. Then for every S 2 L, 2 H(!1 ) there is some T  S such thatT

L ẋ is equivalent to a Mathias generic over L[, S]

Mutatis mutandis...
Remark: We proved also that the lemma above holds by replacing S, U and ML
instead of L.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Other sharps

Let n < !, x 2 R. Mn# (x) is the minimal countable active
mouse constructed from x that contains n Woodin
cardinals.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Other sharps

Let n < !, x 2 R. Mn# (x) is the minimal countable active
mouse constructed from x that contains n Woodin
cardinals.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Theorem (C.-S.)
Suppose P 2 {S, M, U, L, ML}. Then P preserves projective determinacy.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Theorem (C.-S.)
Suppose P 2 {S, M, U, L, ML}. Then P preserves projective determinacy.
By a result of Martin-Neeman-Steel-Woodin, projective determinacy is equivalent
to the existence of Mn# (x) for every n < !, x 2 R. We prove this theorem by
induction on n and it is a li�le technical but we use strongly the suitable version of
our star lemma.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Some applications

If PD holds in V , forcing with some of the tree forcings above does not add
new equivalence classes to thin equivalence relations on the reals.

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Some applications

If PD holds in V , forcing with some of the tree forcings above does not add
new equivalence classes to thin equivalence relations on the reals.
Suppose every real has a sharp. If P is some of the tree forcing notions
mentioned then the value of the second uniform indiscernible
u2 = sup{(!1 )+L[x] | x 2 ! 2}P

does not change, i.e. u2V = u2V .

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster

Motivation

Sharps for reals

Sacks forcing

Silver forcing

Mathias forcing

Laver forcing#
About Mn (x)Applications

Thank you for your a�ention!

Fabiana Castiblanco
Preserving sharps by tree forcings

Universität Münster