syntactical and semantical approaches to generic multiverse
TRANSCRIPT
Syntactical and Semantical approaches to Generic
Multiverse
Toshimichi Usuba(薄葉 季路)
Waseda University
December 7 , 2018
Sendai Logic School 2018
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 1 / 39
Generic Multivserse
In 1980’s, Woodin introduced a concept of Generic Multiverse:
Definition
A collection M is a Generic Multiverse (GM) if :
1 M is a family of models of ZFC.
2 M is closed under taking ground models and generic extensions.
3 For every universe M,N ∈ M, M is connected with N.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 2 / 39
Woodin’s generic multiverse
Definition (Woodin 1980’s)
A collection M is a Woodin’s generic multiverse if :
1 M is a family of countable transitive ∈-models of ZFC.
2 M is closed under taking ground models; for every M ∈ M, if
N ⊆ M is a ground model of M then N ∈ M.
3 For every M ∈ M, poset P ∈ M, and (M,P)-generic G , we have
M[G ] ∈ M.
4 For every M,N ∈ M, there are finitely many M0, . . . ,Mn ∈ M such
that M0 = M, Mn = N, and each Mi is a ground model or a generic
extension of Mi+1.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 3 / 39
Woodin’s Generic multiverse can be seen as Kripke frame; a family of
possible worlds.
In the view point of multiverse conception, one can say that:
The Continuum Hypothesis is neither TRUE nor FALSE, because there
are two worlds M,N ∈ M such that CH is true in M but false in N.
However, (under certain Large Cardinal Axiom), the regularity
properties of the projective sets of the reals is TRUE, because it is true
in any worlds of M.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 4 / 39
Woodin’s Generic multiverse can be seen as Kripke frame; a family of
possible worlds.
In the view point of multiverse conception, one can say that:
The Continuum Hypothesis is neither TRUE nor FALSE, because there
are two worlds M,N ∈ M such that CH is true in M but false in N.
However, (under certain Large Cardinal Axiom), the regularity
properties of the projective sets of the reals is TRUE, because it is true
in any worlds of M.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 4 / 39
Remark
Let M be a Woodin’s generic multiverse.
1 Since each M ∈ M is countable, M satisfies: For every M ∈ M and
P ∈ M, there is an (M,P)-generic G with M[G ] ∈ M.
2 For a given countable transitive ∈-model M of ZFC, there is a unique
Woodin’s generic multiverse M with M ∈ M. In this sense, M is the
generic multiverse containing M, or the generic multiverse generated
by M.
The construction of Woodin’s GM1 Fix a countable transitive ∈-model M of ZFC.
2 Let M0 = M, and Mn+1 be the set of all N which is a ground
model or a generic extension of some W ∈ Mn.
3 M =∪
n<ω Mn is the generic multiverse generated by M.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 5 / 39
Remark
Let M be a Woodin’s generic multiverse.
1 Since each M ∈ M is countable, M satisfies: For every M ∈ M and
P ∈ M, there is an (M,P)-generic G with M[G ] ∈ M.
2 For a given countable transitive ∈-model M of ZFC, there is a unique
Woodin’s generic multiverse M with M ∈ M. In this sense, M is the
generic multiverse containing M, or the generic multiverse generated
by M.
The construction of Woodin’s GM1 Fix a countable transitive ∈-model M of ZFC.
2 Let M0 = M, and Mn+1 be the set of all N which is a ground
model or a generic extension of some W ∈ Mn.
3 M =∪
n<ω Mn is the generic multiverse generated by M.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 5 / 39
Steel’s generic multiverse
Definition (Steel 2014)
A collection M is a Steel’s generic multiverse if :
1 M is a family of transitive ∈-models (not necessary countable, nor
set) of ZFC.
2 M is closed under taking ground models.
3 For every M ∈ M and poset P ∈ M, there is an (M,P)-generic G
with M[G ] ∈ M.
4 (Amalgamation) For every M,N ∈ M there is W ∈ M such that W
is a common generic extension of M and N.
Amalgamation represents “Every world has the same information”.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 6 / 39
Standard construction of Steel’s generic multiverse
1 Fix a countable transitive ∈-model M of ZFC.
2 Consider Coll(< ON) in M; it is a class forcing notion which adds a
surjection from ω onto each ordinals in M.
3 Take an (M,Coll(< ON))-generic G .
4 Let M = N | N is a ground model of M[Gα] for some α, whereGα = G ∩ Coll(α) is generic adding surjection from ω onto α.
5 M satisfies the conditions of Steel’s generic multiverse.
Remark
Unlike Woodin’s generic multiverse, this construction depends on the
choice of an (M,Coll(< ON))-generic G .
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 7 / 39
Standard construction of Steel’s generic multiverse
1 Fix a countable transitive ∈-model M of ZFC.
2 Consider Coll(< ON) in M; it is a class forcing notion which adds a
surjection from ω onto each ordinals in M.
3 Take an (M,Coll(< ON))-generic G .
4 Let M = N | N is a ground model of M[Gα] for some α, whereGα = G ∩ Coll(α) is generic adding surjection from ω onto α.
5 M satisfies the conditions of Steel’s generic multiverse.
Remark
Unlike Woodin’s generic multiverse, this construction depends on the
choice of an (M,Coll(< ON))-generic G .
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 7 / 39
Both Woodin and Steel’s generic multiverses need ZFC as a
background theory.
We want to develop the theory of generic multiverse without
background ZFC.
Question
Can we develop a formal system, or axiomatization of Generic Multiverse?
For instance, is there a first order (or some nice) theory T which
axiomatizes (Woodin or Steel’s) generic multiverse in the sense of Model
theory?
Vaananen developed multiverse logic using his dependence logic, and
observed Steel’s GM.
Steel gave such a theory which characterize his GM.
Steel’s approach is more direct, so in this talk we will consider a
variant of his approach.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 8 / 39
Both Woodin and Steel’s generic multiverses need ZFC as a
background theory.
We want to develop the theory of generic multiverse without
background ZFC.
Question
Can we develop a formal system, or axiomatization of Generic Multiverse?
For instance, is there a first order (or some nice) theory T which
axiomatizes (Woodin or Steel’s) generic multiverse in the sense of Model
theory?
Vaananen developed multiverse logic using his dependence logic, and
observed Steel’s GM.
Steel gave such a theory which characterize his GM.
Steel’s approach is more direct, so in this talk we will consider a
variant of his approach.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 8 / 39
Both Woodin and Steel’s generic multiverses need ZFC as a
background theory.
We want to develop the theory of generic multiverse without
background ZFC.
Question
Can we develop a formal system, or axiomatization of Generic Multiverse?
For instance, is there a first order (or some nice) theory T which
axiomatizes (Woodin or Steel’s) generic multiverse in the sense of Model
theory?
Vaananen developed multiverse logic using his dependence logic, and
observed Steel’s GM.
Steel gave such a theory which characterize his GM.
Steel’s approach is more direct, so in this talk we will consider a
variant of his approach.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 8 / 39
Generic Multiverse logic GM
GM is a first order two-sorted logic:
set (first order) variables: x , y , z , . . .
world (class, second-order) variables: M,N,W , . . . .
predicate symbols: ∈, =.
Atomic formulas: x = y , M = N, x ∈ y , x ∈ M.
A GM-structure will be of the form M = (SM,WM,∈M);
the set-part SM; a family of possible sets.
the world-part W ⊆ P(SM); a family of possible worlds.
We define ⊨ and ⊢ by standard ways.
A formula (sentence) of set-theory is a formula (sentence) of GMwhich does not contain world variables.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 9 / 39
Axioms of Steel’s generic multiverse
TS consists of:
1 ∀M(σM) where σ ∈ ZFC.
2 ∀M∀x ∈ M∀y ∈ x(y ∈ M), ∀x∃M(x ∈ M)
3 ∀M∀N(M = N ↔ ∀x(x ∈ M ↔ x ∈ N)).
4 Closed under taking ground models;
For every M ∈ M and ground N of M, we have N ∈ M.
5 Closed under taking generic extensions;
For every M ∈ M and poset P ∈ M, there is an (M,P)-generic G with
M[G ] ∈ M.
6 Amalgamation.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 10 / 39
Definability of ground models
How to describe “closed under taking ground models” in the language
of GM?
Fact (Laver, Woodin, Fuchs-Hamkins-Reitz, in ZFC or NBG)
There is a formula φG (x , y) of set-theory such that:
1 For every r ∈ V , the definable class Wr = x | φG (x , r) is a
transitive model of ZFC and is a ground model of V .
2 For every transitive model W ⊆ V of ZFC, if W is a ground model of
V then W = Wr for some r ∈ W .
So “closed under taking ground models” can be:
∀M∀r ∈ M∃N(N = x ∈ M | φG (x , r)M).
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 11 / 39
Definability of ground models
How to describe “closed under taking ground models” in the language
of GM?
Fact (Laver, Woodin, Fuchs-Hamkins-Reitz, in ZFC or NBG)
There is a formula φG (x , y) of set-theory such that:
1 For every r ∈ V , the definable class Wr = x | φG (x , r) is a
transitive model of ZFC and is a ground model of V .
2 For every transitive model W ⊆ V of ZFC, if W is a ground model of
V then W = Wr for some r ∈ W .
So “closed under taking ground models” can be:
∀M∀r ∈ M∃N(N = x ∈ M | φG (x , r)M).
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 11 / 39
Axiom of Steel’s GM TS
1 ∀M(σM) (where σ ∈ ZFC).
2 ∀M∀x ∈ M∀y ∈ x(y ∈ M), ∀x∃M(x ∈ M).
3 ∀M∀N(M = N ↔ ∀x(x ∈ M ↔ x ∈ N)).
4 ∀M∀r ∈ M∃N(N = x ∈ M | φG (x , r)M)
5 ∀M∀P ∈ M∃g∃N(g is (M,P)-generic ∧ N = M[g ]).
6 ∀M∀N∃W (M and N are ground models of W ).
Where
g is (M,P)-generic ⇐⇒ g ⊆ P is a filter ∧∀D ∈ M(D is dense in
P → D ∩ g = ∅).
N = M[g ] ⇐⇒ P, g ∈ N ∧ ∀x ∈ N∃τ ∈ M(τ is P-name∧τg = x).
It is easy to check that M ⊨ TS ⇐⇒ the world part of M is a Steel’s
generic multiverse.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 12 / 39
Can Woodin’s generic multiverse be axiomatized by a similar way?
Let us forget the condition “countable transitive model”.
Countability may need to only guarantee that: every world has a
generic extension.
The axiom TW would be:
1 ∀M(σM) (where σ ∈ ZFC).
2 ∀M∀x ∈ M∀y ∈ x(y ∈ M), ∀x∃M(x ∈ M).
3 ∀M∀N(M = N ↔ ∀x(x ∈ M ↔ x ∈ N)).
4 ∀M∀r ∈ M∃N(N = x ∈ M | φG (x , r)M)
5 ∀M∀P ∈ M∃g∃N(g is (M,P)-generic ∧ N = M[g ]).
6 For every M ∈ M, poset P ∈ M, and (M,P)-generic G , we have
M[G ] ∈ M.
7 Each M ∈ M is connected with other N ∈ M by finite paths.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 13 / 39
Can Woodin’s generic multiverse be axiomatized by a similar way?
Let us forget the condition “countable transitive model”.
Countability may need to only guarantee that: every world has a
generic extension.
The axiom TW would be:
1 ∀M(σM) (where σ ∈ ZFC).
2 ∀M∀x ∈ M∀y ∈ x(y ∈ M), ∀x∃M(x ∈ M).
3 ∀M∀N(M = N ↔ ∀x(x ∈ M ↔ x ∈ N)).
4 ∀M∀r ∈ M∃N(N = x ∈ M | φG (x , r)M)
5 ∀M∀P ∈ M∃g∃N(g is (M,P)-generic ∧ N = M[g ]).
6 For every M ∈ M, poset P ∈ M, and (M,P)-generic G , we have
M[G ] ∈ M.
7 Each M ∈ M is connected with other N ∈ M by finite paths.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 13 / 39
6 For every M ∈ M, poset P ∈ M, (M,P)-generic G , we have
M[G ] ∈ M.
This may be interpreted as:
Strong Closure
∀M∀P ∈ M∀g(g is (M,P)-generic → ∃N(N = M[g ])).
But this is not sufficient: generic filter g ranges only over the set-part of a
model, but there might be a generic filter outside of a model.
7 Each M ∈ M is connected with other N ∈ M by finite paths.
This is a serious problem: “finite paths” can not be described by our logic!
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 14 / 39
6 For every M ∈ M, poset P ∈ M, (M,P)-generic G , we have
M[G ] ∈ M.
This may be interpreted as:
Strong Closure
∀M∀P ∈ M∀g(g is (M,P)-generic → ∃N(N = M[g ])).
But this is not sufficient: generic filter g ranges only over the set-part of a
model, but there might be a generic filter outside of a model.
7 Each M ∈ M is connected with other N ∈ M by finite paths.
This is a serious problem: “finite paths” can not be described by our logic!
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 14 / 39
6 For every M ∈ M, poset P ∈ M, (M,P)-generic G , we have
M[G ] ∈ M.
This may be interpreted as:
Strong Closure
∀M∀P ∈ M∀g(g is (M,P)-generic → ∃N(N = M[g ])).
But this is not sufficient: generic filter g ranges only over the set-part of a
model, but there might be a generic filter outside of a model.
7 Each M ∈ M is connected with other N ∈ M by finite paths.
This is a serious problem: “finite paths” can not be described by our logic!
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 14 / 39
6 For every M ∈ M, poset P ∈ M, (M,P)-generic G , we have
M[G ] ∈ M.
This may be interpreted as:
Strong Closure
∀M∀P ∈ M∀g(g is (M,P)-generic → ∃N(N = M[g ])).
But this is not sufficient: generic filter g ranges only over the set-part of a
model, but there might be a generic filter outside of a model.
7 Each M ∈ M is connected with other N ∈ M by finite paths.
This is a serious problem: “finite paths” can not be described by our logic!
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 14 / 39
The downward directedness of grounds
The downward directedness of grounds helps this problem.
Theorem (Usuba, downward directedness of grounds, in ZFC)
For every models M and N of ZFC, if M and N are ground models of some
supermodel, then M and N has a common ground model W of M and N.
Corollary
If M is a Woodin or Steel’s generic multiverse, then for every M,N ∈ Mthere is W which is a common ground model of M and N.
If M has the amalgamation property, it is immediate.
If M is a Woodin’s GM, then it follows from the construction.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 15 / 39
The downward directedness of grounds
The downward directedness of grounds helps this problem.
Theorem (Usuba, downward directedness of grounds, in ZFC)
For every models M and N of ZFC, if M and N are ground models of some
supermodel, then M and N has a common ground model W of M and N.
Corollary
If M is a Woodin or Steel’s generic multiverse, then for every M,N ∈ Mthere is W which is a common ground model of M and N.
If M has the amalgamation property, it is immediate.
If M is a Woodin’s GM, then it follows from the construction.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 15 / 39
The downward directedness of grounds
The downward directedness of grounds helps this problem.
Theorem (Usuba, downward directedness of grounds, in ZFC)
For every models M and N of ZFC, if M and N are ground models of some
supermodel, then M and N has a common ground model W of M and N.
Corollary
If M is a Woodin or Steel’s generic multiverse, then for every M,N ∈ Mthere is W which is a common ground model of M and N.
If M has the amalgamation property, it is immediate.
If M is a Woodin’s GM, then it follows from the construction.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 15 / 39
The downward directedness of grounds
The downward directedness of grounds helps this problem.
Theorem (Usuba, downward directedness of grounds, in ZFC)
For every models M and N of ZFC, if M and N are ground models of some
supermodel, then M and N has a common ground model W of M and N.
Corollary
If M is a Woodin or Steel’s generic multiverse, then for every M,N ∈ Mthere is W which is a common ground model of M and N.
If M has the amalgamation property, it is immediate.
If M is a Woodin’s GM, then it follows from the construction.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 15 / 39
7 Each M ∈ M is connected with other N ∈ M by finite paths.
is equivalent to
Every M,N ∈ M have a common ground model.
This can be described by:
∀M∀N∃W (W is a ground model of M and N).
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 16 / 39
7 Each M ∈ M is connected with other N ∈ M by finite paths.
is equivalent to
Every M,N ∈ M have a common ground model.
This can be described by:
∀M∀N∃W (W is a ground model of M and N).
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 16 / 39
Axioms of Woodin’s GM TW (approximation)
1 ∀M(σM) (where σ ∈ ZFC).
2 ∀M∀x ∈ M∀y ∈ x(y ∈ M), ∀x∃M(x ∈ M).
3 ∀M∀N(M = N ↔ ∀x(x ∈ M ↔ x ∈ N)).
4 ∀M∀r ∈ M∃N(N = x ∈ M | φG (x , r)M)
5 ∀M∀P ∈ M∃g∃N(g is (M,P)-generic ∧ N = M[g ]).
6 ∀M∀P ∈ M∀g(g is (M,P)-generic → ∃N(N = M[g ])).
7 ∀M∀N∃W (W is a ground model of M and N).
Woodin’s generic multiverse is a model of TW .
However the converse does not hold.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 17 / 39
The basic GM
Let us consider the intersection of TS and TW , which grabs the essence of
Generic Multiverse.
Axioms of basic GM, TGM
1 ∀M(σM) (where σ ∈ ZFC).
2 ∀M∀x ∈ M∀y ∈ x(y ∈ M), ∀x∃M(x ∈ M).
3 ∀M∀N(M = N ↔ ∀x(x ∈ M ↔ x ∈ N)).
4 ∀M∀r ∈ M∃N(N = x ∈ M | φG (x , r)M)
5 ∀M∀P∃g∃N(g is (M,P)-generic ∧ N = M[g ]).
6 ∀M∀N∃W (W is a ground model of M and N).
TW = TGM+Strong Closure: ∀M∀P∀g(g is (M,P)-generic→ ∃N(N = M[g ])).
TS = TGM+Amalgamation.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 18 / 39
Convention
For a model M of TGM , x ∈ M means that x is an element of the
set-part of M, and M ∈ M means that M is of the world-part. For
x , y ,M ∈ M, x ∈ y means M ⊨ x ∈ y , and x ∈ M does M ⊨ x ∈ M.
Lemma (Forcing Theorem)
For every model M of TGM , M ∈ M, P ∈ M, and sentence σ of
set-theory, the following are equivalent:
1 M ⊨ (⊩P σ)M .
2 M ⊨ ∀N∀g(g is (M,P)-generic ∧N = M[g ] → σN).
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 19 / 39
Convention
For a model M of TGM , x ∈ M means that x is an element of the
set-part of M, and M ∈ M means that M is of the world-part. For
x , y ,M ∈ M, x ∈ y means M ⊨ x ∈ y , and x ∈ M does M ⊨ x ∈ M.
Lemma (Forcing Theorem)
For every model M of TGM , M ∈ M, P ∈ M, and sentence σ of
set-theory, the following are equivalent:
1 M ⊨ (⊩P σ)M .
2 M ⊨ ∀N∀g(g is (M,P)-generic ∧N = M[g ] → σN).
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 19 / 39
Definition (Woodin)
A sentence σ of set-theory is a multiverse truth if σ is true in any world of
a generic multiverse.
Fact (Woodin)
There is a computable translation σ → σ∗ such that for every Woodin’s
GM M, σ is a multiverse truth (in M)
⇐⇒ M ⊨ σ∗ for some M ∈ M⇐⇒ M ⊨ σ∗ for every M ∈ M.
Lemma
There is a computable translation σ → σ∗ such that for every model M of
TGS , σ is a multiverse truth (in M)
⇐⇒ M ⊨ ∃M((σ∗)M).
⇐⇒ M ⊨ ∀M((σ∗)M).
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 20 / 39
Definition (Woodin)
A sentence σ of set-theory is a multiverse truth if σ is true in any world of
a generic multiverse.
Fact (Woodin)
There is a computable translation σ → σ∗ such that for every Woodin’s
GM M, σ is a multiverse truth (in M)
⇐⇒ M ⊨ σ∗ for some M ∈ M⇐⇒ M ⊨ σ∗ for every M ∈ M.
Lemma
There is a computable translation σ → σ∗ such that for every model M of
TGS , σ is a multiverse truth (in M)
⇐⇒ M ⊨ ∃M((σ∗)M).
⇐⇒ M ⊨ ∀M((σ∗)M).
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 20 / 39
Definition (Woodin)
A sentence σ of set-theory is a multiverse truth if σ is true in any world of
a generic multiverse.
Fact (Woodin)
There is a computable translation σ → σ∗ such that for every Woodin’s
GM M, σ is a multiverse truth (in M)
⇐⇒ M ⊨ σ∗ for some M ∈ M⇐⇒ M ⊨ σ∗ for every M ∈ M.
Lemma
There is a computable translation σ → σ∗ such that for every model M of
TGS , σ is a multiverse truth (in M)
⇐⇒ M ⊨ ∃M((σ∗)M).
⇐⇒ M ⊨ ∀M((σ∗)M).
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 20 / 39
General question
Let M be a model of TGM . What is the structure of the set-part of M?
Fact (Mostowski, Woodin)
Let M be a countable transitive ∈-model of ZFC. Then M has two generic
extensions N0, N1 such that there is no common generic extension of N0
and N1.
Corollary
Let M be a Woodin’s generic multiverse.
1 M cannot satisfy Amalgamation.
2 Indeed the set-part of M does not satisfy the paring axiom
∀x∀y∃z(z = x , y). So TGM + ¬Paring is consistent.
Under TGM , Paring ⇐⇒ ∀x∀y∃M(x , y ∈ M).
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 21 / 39
General question
Let M be a model of TGM . What is the structure of the set-part of M?
Fact (Mostowski, Woodin)
Let M be a countable transitive ∈-model of ZFC. Then M has two generic
extensions N0, N1 such that there is no common generic extension of N0
and N1.
Corollary
Let M be a Woodin’s generic multiverse.
1 M cannot satisfy Amalgamation.
2 Indeed the set-part of M does not satisfy the paring axiom
∀x∀y∃z(z = x , y). So TGM + ¬Paring is consistent.
Under TGM , Paring ⇐⇒ ∀x∀y∃M(x , y ∈ M).
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 21 / 39
General question
Let M be a model of TGM . What is the structure of the set-part of M?
Fact (Mostowski, Woodin)
Let M be a countable transitive ∈-model of ZFC. Then M has two generic
extensions N0, N1 such that there is no common generic extension of N0
and N1.
Corollary
Let M be a Woodin’s generic multiverse.
1 M cannot satisfy Amalgamation.
2 Indeed the set-part of M does not satisfy the paring axiom
∀x∀y∃z(z = x , y). So TGM + ¬Paring is consistent.
Under TGM , Paring ⇐⇒ ∀x∀y∃M(x , y ∈ M).
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 21 / 39
In the standard construction of Steel’s GM, the set-part is a class forcing
extension of some world via Coll(< ON).
Fact
The class forcing Coll(< ON) forces ZFC−Power set Axiom+“every set is
countable”.
Lemma
Let M be a Steel’s generic multiverse by the standard construction. Then
the set-part of M is a model of ZFC−Power set Axiom+“every set is
countable”.
Question
What is the structure of the set-part of a model of TGM+Amalgamation?
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 22 / 39
In the standard construction of Steel’s GM, the set-part is a class forcing
extension of some world via Coll(< ON).
Fact
The class forcing Coll(< ON) forces ZFC−Power set Axiom+“every set is
countable”.
Lemma
Let M be a Steel’s generic multiverse by the standard construction. Then
the set-part of M is a model of ZFC−Power set Axiom+“every set is
countable”.
Question
What is the structure of the set-part of a model of TGM+Amalgamation?
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 22 / 39
In the standard construction of Steel’s GM, the set-part is a class forcing
extension of some world via Coll(< ON).
Fact
The class forcing Coll(< ON) forces ZFC−Power set Axiom+“every set is
countable”.
Lemma
Let M be a Steel’s generic multiverse by the standard construction. Then
the set-part of M is a model of ZFC−Power set Axiom+“every set is
countable”.
Question
What is the structure of the set-part of a model of TGM+Amalgamation?
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 22 / 39
In the standard construction of Steel’s GM, the set-part is a class forcing
extension of some world via Coll(< ON).
Fact
The class forcing Coll(< ON) forces ZFC−Power set Axiom+“every set is
countable”.
Lemma
Let M be a Steel’s generic multiverse by the standard construction. Then
the set-part of M is a model of ZFC−Power set Axiom+“every set is
countable”.
Question
What is the structure of the set-part of a model of TGM+Amalgamation?
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 22 / 39
Theorem
Let M be a model of TGM . If the set-part of M satisfies Paring, then for
every x0, . . . , xn ∈ M and formula φ of set-theory, the following are
equivalent:
1 The set-part of M satisfies φ(x0, . . . , xn).
2 For every M ∈ M,
M ⊨ x0, . . . , xn ∈ M → (⊩Coll(<ON) φ(x0, . . . , xn))M .
3 There exists M ∈ M such that
M ⊨ x0, . . . , xn ∈ M ∧ (⊩Coll(<ON) φ(x0, . . . , xn))M .
Roughly speaking, if M satisfies Paring, then each world knows the truths
of the set-part.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 23 / 39
Corollary
Let M be a model of TGM .
1 If the set-part of M satisfies Paring, then it is a model of
ZFC−Power set Axiom+“every set is countable”:
TGM + Paring ⊢ ZFC−Power set+“every set is countable”.
2 In particular TGM+Amalgamation⊢ ZFC−Power set+“every set is
countable”.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 24 / 39
Sketch of the proof
By induction on the complexity of the formula φ.
If φ is atomic, it is clear.
The boolean combination step is easy.
Suppose φ = ∃xψ(x). If Coll(< ON) forces φ over M, then there is α and a generic extension
M[g ] ∈ M of M via Coll(α), and y ∈ M[g ] such that Coll(< ON)
forces ψ(y) over M[g ]. Then ψ(y) holds in the set-part of M.
If φ holds in the set-part of M, pick a witness z ∈ M.
Choose N ∈ M with z ∈ N, and W ∈ M which is a common ground
model of M and N.
Then Coll(< ON) forces φ over W , and so does over M.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 25 / 39
Amalgamation is a Π12-statement.
Paring is a Π02-statement, so the complexity is drastically reduced.
Question
Is TGM+Paring equivalent to TS?
TGM+Paring⊢Amalgamation?
Theorem
It is consistent that TGM+Paring+¬Amalgamation.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 26 / 39
Amalgamation is a Π12-statement.
Paring is a Π02-statement, so the complexity is drastically reduced.
Question
Is TGM+Paring equivalent to TS?
TGM+Paring⊢Amalgamation?
Theorem
It is consistent that TGM+Paring+¬Amalgamation.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 26 / 39
Amalgamation is a Π12-statement.
Paring is a Π02-statement, so the complexity is drastically reduced.
Question
Is TGM+Paring equivalent to TS?
TGM+Paring⊢Amalgamation?
Theorem
It is consistent that TGM+Paring+¬Amalgamation.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 26 / 39
Minimum ground
Fact (Fuchs-Hamkins-Reitz)
1 It is consistent that ZFC+“there exists the minimum ground model”.
2 It is consistent that ZFC+Large Cardinal Axiom+“there exists the
minimum ground model”.
3 It is consistent that ZFC+“there is no minimal ground models”.
Remark
By the definability of all ground models, the statement “there exists the
minimum ground model” is a sentence of set-theory.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 27 / 39
Theorem
Let M be a model of TGS . Then the following are equivalent:
1 M has the minimum world.
2 There is M ∈ M such that “there exists the minimum ground model”
holds in M.
3 For every M ∈ M, “there exists the minimum ground model” holds in
M.
Actually, for M ∈ M, if the intersection of all ground models of M is a
ground model of M, then it is the minimum world of M.
In particular, there is a sentence of set-theory σ such that
M has the minimum world
⇐⇒ M ⊨ ∃N∀M(N ⊆ M) ⇐⇒ M ⊨ ∃M(σM) ⇐⇒ M ⊨ ∀M(σM).
So each world knows whether there is the minimum world or not.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 28 / 39
Proposition
TGM+Paring+“there exists the minimum world” ⊢Amalgamation.
Hence under TGM+“there exists the minimum world”, Pairing is
equivalent to Amalgamation.
Proof.1 Let W0 ∈ M be the minimum world.
2 Take M,N ∈ M. Then M = W0[g ] and N = W0[h] for some generic
g , h ∈ M.
3 By Pairing, there is W ∈ M with g , h ∈ W .
4 Then W0 ⊆ W and g , h ∈ W , so M,N ⊆ W .
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 29 / 39
Large cardinal
Theorem (Usuba, in ZFC)
If there exists an extendible cardinal, then V has the minimum ground
model.
An uncountable cardinal κ is extendible if for every α ≥ κ, there are β and
an elementary embedding j : Vα → Vβ with critical point κ and α < j(κ).
Corollary
TGM+Paring+“some world has a large cardinal” ⊢Amalgamation.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 30 / 39
Large cardinal
Theorem (Usuba, in ZFC)
If there exists an extendible cardinal, then V has the minimum ground
model.
An uncountable cardinal κ is extendible if for every α ≥ κ, there are β and
an elementary embedding j : Vα → Vβ with critical point κ and α < j(κ).
Corollary
TGM+Paring+“some world has a large cardinal” ⊢Amalgamation.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 30 / 39
Sketch of a construction of TGM+Paring+¬Amalgamation
1 Fix a countable transitive ∈-model M of ZFC.
2 Take a class Easton forcing extension M[H] of M.
3 It is known that M[H] has no minimal ground models.
4 Do the standard construction of Steel’s GM starting from M[H].
5 Let M be a Steel’s GM.
6 Remove some worlds from M carefully.
7 By careful removing, we can get a model N of
TGM+Paring+¬Amalgamation.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 31 / 39
Our example N is a sub-multiverse of some model of TS .
Definition
Let M = (S,W) and M′ = (S ′,W ′) be models of TGM . M′ is a
worlds-extension of M if S = S ′ and W ⊆ W ′.
Question (Open)
Does every model of TGM+Paring have a worlds-extension which is a
model of TGM+Amalgamation?
Remark
TGM+Paring+¬Strong Closure is consistent; N above is a witness.
Proposition
Let M be a model of TGS . If M satisfies Paring, then M has a
worlds-extension satisfying Strong Closure+Paring.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 32 / 39
Our example N is a sub-multiverse of some model of TS .
Definition
Let M = (S,W) and M′ = (S ′,W ′) be models of TGM . M′ is a
worlds-extension of M if S = S ′ and W ⊆ W ′.
Question (Open)
Does every model of TGM+Paring have a worlds-extension which is a
model of TGM+Amalgamation?
Remark
TGM+Paring+¬Strong Closure is consistent; N above is a witness.
Proposition
Let M be a model of TGS . If M satisfies Paring, then M has a
worlds-extension satisfying Strong Closure+Paring.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 32 / 39
Our example N is a sub-multiverse of some model of TS .
Definition
Let M = (S,W) and M′ = (S ′,W ′) be models of TGM . M′ is a
worlds-extension of M if S = S ′ and W ⊆ W ′.
Question (Open)
Does every model of TGM+Paring have a worlds-extension which is a
model of TGM+Amalgamation?
Remark
TGM+Paring+¬Strong Closure is consistent; N above is a witness.
Proposition
Let M be a model of TGS . If M satisfies Paring, then M has a
worlds-extension satisfying Strong Closure+Paring.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 32 / 39
Our example N is a sub-multiverse of some model of TS .
Definition
Let M = (S,W) and M′ = (S ′,W ′) be models of TGM . M′ is a
worlds-extension of M if S = S ′ and W ⊆ W ′.
Question (Open)
Does every model of TGM+Paring have a worlds-extension which is a
model of TGM+Amalgamation?
Remark
TGM+Paring+¬Strong Closure is consistent; N above is a witness.
Proposition
Let M be a model of TGS . If M satisfies Paring, then M has a
worlds-extension satisfying Strong Closure+Paring.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 32 / 39
1 Note that since M is a model of TGS+Paring, the set-part of M is a
model of ZFC−Power set.
2 Hence for every M ∈ M, P ∈ M, and (M,P)-generic g ∈ M, M[g ] is
(second-order) definable class in M, and by forcing theorem, M[g ] is
a model of ZFC.
3 Just let W ′ = M[g ] | M ∈ M, g ∈ M is (M,P)-generic for some
P ∈ M, and M′ = (S,W ′).
4 Point is: for every M ∈ M, α ∈ M, (M,Coll(α))-generic g , and
β ∈ M, there is g ′ ∈ M which is (M, coll(β))-generic and M[g ] is a
ground model of M[g ′].
5 This guarantees that M′ = (S,W ′) is a model of
TGS+Paring+Strong Closure.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 33 / 39
Remark1 For a model M be a model of TGM+Paring, let M′ = (S,W ′) be a
world extension of M constructed as above. Then W ′ is
(second-order) definable in M.
2 In this sense, the worlds-extension M′ is a very weak extension of M;
There is a computable translation σ 7→ σ∗ for sentences σ in GM such
that M′ ⊨ σ ⇐⇒ M ⊨ σ∗. So the truth of M′ is definable in M.
3 In particular
TGM+Paring+Strong Closure⊢ σ ⇐⇒ TGM+Paring⊢ σ∗.
If we strengthen Paring to Amalgamation, we have:
Lemma
TGM+Amalgamation⊢Strong Closure.
In particular, every model of TS is a model of TW .
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 34 / 39
Remark1 For a model M be a model of TGM+Paring, let M′ = (S,W ′) be a
world extension of M constructed as above. Then W ′ is
(second-order) definable in M.
2 In this sense, the worlds-extension M′ is a very weak extension of M;
There is a computable translation σ 7→ σ∗ for sentences σ in GM such
that M′ ⊨ σ ⇐⇒ M ⊨ σ∗. So the truth of M′ is definable in M.
3 In particular
TGM+Paring+Strong Closure⊢ σ ⇐⇒ TGM+Paring⊢ σ∗.
If we strengthen Paring to Amalgamation, we have:
Lemma
TGM+Amalgamation⊢Strong Closure.
In particular, every model of TS is a model of TW .
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 34 / 39
Definability of worlds
In ZFC, every ground model is definable.
In TGM+Paring, the set-part is close to a class forcing extension of
each world.
Question
Let M be a model of TGM+Paring. Is each world definable in the set-part
by a formula of set-theory? That is, is there a formula φ of set-theory such
that for every M ∈ M,
M ⊨ M = x ∈ SM | φ(x , r)
for some r ∈ M?
It is true if M has a world of V = L.
If this is possible, then the world-part is definable in the set-part, so it
become a redundant part...Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 35 / 39
Definability of worlds
In ZFC, every ground model is definable.
In TGM+Paring, the set-part is close to a class forcing extension of
each world.
Question
Let M be a model of TGM+Paring. Is each world definable in the set-part
by a formula of set-theory? That is, is there a formula φ of set-theory such
that for every M ∈ M,
M ⊨ M = x ∈ SM | φ(x , r)
for some r ∈ M?
It is true if M has a world of V = L.
If this is possible, then the world-part is definable in the set-part, so it
become a redundant part...Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 35 / 39
Definability of worlds
In ZFC, every ground model is definable.
In TGM+Paring, the set-part is close to a class forcing extension of
each world.
Question
Let M be a model of TGM+Paring. Is each world definable in the set-part
by a formula of set-theory? That is, is there a formula φ of set-theory such
that for every M ∈ M,
M ⊨ M = x ∈ SM | φ(x , r)
for some r ∈ M?
It is true if M has a world of V = L.
If this is possible, then the world-part is definable in the set-part, so it
become a redundant part...Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 35 / 39
Theorem
There is a model M of TGM+Amalgamation and M ∈ M such that for
every formula φ of set-theory and r ∈ M, we have
M ⊨ M = x ∈ SM | φ(x , r).
Hence M is never definable in M by a formula of set-theory.
However our model M above has no minimal worlds.
Question (Open)
Suppose M is a model of TGM+Amalgamation. If M has the minimum
world, then is the minimum world definable in M by a formula of
set-theory?
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 36 / 39
Theorem
There is a model M of TGM+Amalgamation and M ∈ M such that for
every formula φ of set-theory and r ∈ M, we have
M ⊨ M = x ∈ SM | φ(x , r).
Hence M is never definable in M by a formula of set-theory.
However our model M above has no minimal worlds.
Question (Open)
Suppose M is a model of TGM+Amalgamation. If M has the minimum
world, then is the minimum world definable in M by a formula of
set-theory?
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 36 / 39
Other questions
Can Woodin’s GM be axiomatized exactly by some formal way?
In TGS+Paring, the set-part is a model of ZFC−Power set, a first
order set-theory. Does TGS+Paring imply the replacement scheme for
the formulas of GM? Namely, for every formula φ of GM, does
TGS+Paring imply
∀M∀x∀y(∀z ∈ y∃!wφ(z ,w , x , M) → ∃v∀z ∈ y∃z ∈ vφ(z ,w , x , M))?
It is known that TGM+Amalgamation implies it.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 37 / 39
Conslusions
Generic Multiverse Logic GM.
Basic Theory of Generic Multiverse TGM , this grabs the essence of
Generic Multiverse.
Paring plays important roles in this context.
The existence of the minimum world (or Large Cardinal Axiom)
simplify the structure of GM.
However the deference between TGM+Paring and TS is not so clear
now.
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 38 / 39
References
G. Fuchs, J. D. Hamkins, J. Reitz, Set-theoretic geology. Ann. Pure Appl. Logic
166 (2015), no. 4, 464–501.
J. Steel, Godel’s program. in: Interpreting Godel Critical Essays, Cambridge
University Press, 2014.
T. Usuba, The downward directed grounds hypothesis and very large cardinals. J.
Math. Logic 17, 1750009 (2017)
T. Usuba, Extendible cardinals and the mantle. To appear in Arch. Math. Logic.
J. Vaananen, Multiverse set theory and absolutely undecidable propositions. in:
Interpreting Godel Critical Essays, Cambridge University Press, 2014.
W. H. Woodin, The continuum hypothesis, the generic-multiverse of sets, and the
Ω-conjecture. Set theory, arithmetic, and foundations of mathematics: theorems,
philosophies, 1–42, Lect. Notes Log., 36, Assoc. Symbol. Logic, La Jolla, CA,
2011.
Thank you for your attention!
Toshimichi Usuba (Waseda Univ.) Generic Multiverse Dec. 6, 2018 39 / 39