introduction to forcing
TRANSCRIPT
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Introduction to Forcing
Boban Velickovic
IMJ-PRG Universite de Paris Diderot
Journees GT CalculabilitesIUT de Fontainebleau, April 28 2015
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Outline
1 A brief history of Set Theory
2 Independence results
3 ForcingGeneralitiesFundamental theorem of forcingExamples
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Outline
1 A brief history of Set Theory
2 Independence results
3 ForcingGeneralitiesFundamental theorem of forcingExamples
![Page 4: Introduction to Forcing](https://reader030.vdocument.in/reader030/viewer/2022012614/619c174445dca568d06c166b/html5/thumbnails/4.jpg)
The work of Cantor
In the second half of the 19th century, german mathematician, GeorgCantor laid the foundations of set theory. He defined, ordinal andcardinal numbers, and developed their arithmetic.
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The work of Cantor
In the second half of the 19th century, german mathematician, GeorgCantor laid the foundations of set theory. He defined, ordinal andcardinal numbers, and developed their arithmetic.
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Cantor’s work provoked a lot of controversy.
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DefinitionLet X and Y be sets. We write X ⪯ Y if there is an injection from Xto Y . We write X ≈ Y if there is a bijection between X et Y .
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Theorem (Cantor - Bernstein)Suppose that X ⪯ Y and Y ⪯X . Then X ≈ Y .
PropositionX is infinite iff X ≈X ∖ {x}, for any x ∈X .
DefinitionX is countable if X ≈ N.
Proposition (Cantor)1 If An is countable, for all n, then ⋃nAn is countable.2 An ≈ A, for any infinite set A and integer n ≥ 1.
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Theorem (Cantor - Bernstein)Suppose that X ⪯ Y and Y ⪯X . Then X ≈ Y .
PropositionX is infinite iff X ≈X ∖ {x}, for any x ∈X .
DefinitionX is countable if X ≈ N.
Proposition (Cantor)1 If An is countable, for all n, then ⋃nAn is countable.2 An ≈ A, for any infinite set A and integer n ≥ 1.
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Theorem (Cantor - Bernstein)Suppose that X ⪯ Y and Y ⪯X . Then X ≈ Y .
PropositionX is infinite iff X ≈X ∖ {x}, for any x ∈X .
DefinitionX is countable if X ≈ N.
Proposition (Cantor)1 If An is countable, for all n, then ⋃nAn is countable.2 An ≈ A, for any infinite set A and integer n ≥ 1.
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Theorem (Cantor - Bernstein)Suppose that X ⪯ Y and Y ⪯X . Then X ≈ Y .
PropositionX is infinite iff X ≈X ∖ {x}, for any x ∈X .
DefinitionX is countable if X ≈ N.
Proposition (Cantor)1 If An is countable, for all n, then ⋃nAn is countable.2 An ≈ A, for any infinite set A and integer n ≥ 1.
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However, there are infinite sets that are not countable. By the famousdiagonal argument we have.
Theorem (Cantor)The set of reals R is uncountable.
Cantor spent the rest of his life trying to prove the following.
Continuum Hypothesis (CH)Let X be an infinite set of reals. Then either X ≈ N or X ≈ R.
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However, there are infinite sets that are not countable. By the famousdiagonal argument we have.
Theorem (Cantor)The set of reals R is uncountable.
Cantor spent the rest of his life trying to prove the following.
Continuum Hypothesis (CH)Let X be an infinite set of reals. Then either X ≈ N or X ≈ R.
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However, there are infinite sets that are not countable. By the famousdiagonal argument we have.
Theorem (Cantor)The set of reals R is uncountable.
Cantor spent the rest of his life trying to prove the following.
Continuum Hypothesis (CH)Let X be an infinite set of reals. Then either X ≈ N or X ≈ R.
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However, there are infinite sets that are not countable. By the famousdiagonal argument we have.
Theorem (Cantor)The set of reals R is uncountable.
Cantor spent the rest of his life trying to prove the following.
Continuum Hypothesis (CH)Let X be an infinite set of reals. Then either X ≈ N or X ≈ R.
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However, there are infinite sets that are not countable. By the famousdiagonal argument we have.
Theorem (Cantor)The set of reals R is uncountable.
Cantor spent the rest of his life trying to prove the following.
Continuum Hypothesis (CH)Let X be an infinite set of reals. Then either X ≈ N or X ≈ R.
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Zermelo-Fraenkel set theory
Following a tumultuous period in the Foundations of Mathematics, inthe early 20th century, Ernst Zermelo and Abraham Fraenkelformulated set theory as a first order theory ZF whose onlynonlogical symbol is ∈. This was later augmented by adding theAxiom of Choice.
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ZFC axioms
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Ordinals
In principle, all of mathematics can be carried out in ZFC. So it isimportant to understand its strengths and limitations. The basicconcept is that of an ordinal, which is a generalization of an integer.
Definition1 A well order on a set X is a total order < on X such that every
nonempty subset of X has a minimal element.2 An ordinal is a set α which is transitive (i.e. if x ∈ y ∈ α thenx ∈ α) and well ordered by ∈.
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Ordinals
In principle, all of mathematics can be carried out in ZFC. So it isimportant to understand its strengths and limitations. The basicconcept is that of an ordinal, which is a generalization of an integer.
Definition1 A well order on a set X is a total order < on X such that every
nonempty subset of X has a minimal element.2 An ordinal is a set α which is transitive (i.e. if x ∈ y ∈ α thenx ∈ α) and well ordered by ∈.
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Ordinals
In principle, all of mathematics can be carried out in ZFC. So it isimportant to understand its strengths and limitations. The basicconcept is that of an ordinal, which is a generalization of an integer.
Definition1 A well order on a set X is a total order < on X such that every
nonempty subset of X has a minimal element.2 An ordinal is a set α which is transitive (i.e. if x ∈ y ∈ α thenx ∈ α) and well ordered by ∈.
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We have:0 ∶= ∅,
1 ∶= {0} = {∅},2 ∶= {0,1} = {∅,{∅}},
3 ∶= {0,1,2} = {∅,{∅},{∅,{∅}}},. . .
ω ∶= {0,1,2,3, . . .},ω + 1 ∶= {0,1,2,3, . . . , ω},
. . .ω ⋅ 2 ∶= ω + ω = {0,1,2,3, . . . , ω, ω + 1, ω + 2, ω + 3, . . .},
. . .ω2 ∶= {0,1, . . . , ω, ω + 1, . . . , ω ⋅ 2, ω ⋅ 2 + 1, . . . , ω ⋅ n,ω ⋅ n + 1, . . .},
. . .
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Definition1 The successor of an ordinal α is the ordinal α + 1 = α ∪ {α}.2 An ordinal α is limit if α > 0 and α is not a successor. The least
limit ordinal is ω.
DefinitionA cardinal is an ordinal α such that α ≉ β, for all β < α
Remark1 All integers are cardinals, as well as ω. The ordinalsω + 1, ω + 2, . . ., ω ⋅ 2, . . ., are not cardinals.
2 The first cardinal > ω is denoted by ω1 or ℵ1, the second ω2 orℵ2, etc.
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Definition1 The successor of an ordinal α is the ordinal α + 1 = α ∪ {α}.2 An ordinal α is limit if α > 0 and α is not a successor. The least
limit ordinal is ω.
DefinitionA cardinal is an ordinal α such that α ≉ β, for all β < α
Remark1 All integers are cardinals, as well as ω. The ordinalsω + 1, ω + 2, . . ., ω ⋅ 2, . . ., are not cardinals.
2 The first cardinal > ω is denoted by ω1 or ℵ1, the second ω2 orℵ2, etc.
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Definition1 The successor of an ordinal α is the ordinal α + 1 = α ∪ {α}.2 An ordinal α is limit if α > 0 and α is not a successor. The least
limit ordinal is ω.
DefinitionA cardinal is an ordinal α such that α ≉ β, for all β < α
Remark1 All integers are cardinals, as well as ω. The ordinalsω + 1, ω + 2, . . ., ω ⋅ 2, . . ., are not cardinals.
2 The first cardinal > ω is denoted by ω1 or ℵ1, the second ω2 orℵ2, etc.
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Definition1 The successor of an ordinal α is the ordinal α + 1 = α ∪ {α}.2 An ordinal α is limit if α > 0 and α is not a successor. The least
limit ordinal is ω.
DefinitionA cardinal is an ordinal α such that α ≉ β, for all β < α
Remark1 All integers are cardinals, as well as ω. The ordinalsω + 1, ω + 2, . . ., ω ⋅ 2, . . ., are not cardinals.
2 The first cardinal > ω is denoted by ω1 or ℵ1, the second ω2 orℵ2, etc.
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Definition1 The successor of an ordinal α is the ordinal α + 1 = α ∪ {α}.2 An ordinal α is limit if α > 0 and α is not a successor. The least
limit ordinal is ω.
DefinitionA cardinal is an ordinal α such that α ≉ β, for all β < α
Remark1 All integers are cardinals, as well as ω. The ordinalsω + 1, ω + 2, . . ., ω ⋅ 2, . . ., are not cardinals.
2 The first cardinal > ω is denoted by ω1 or ℵ1, the second ω2 orℵ2, etc.
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Von Neumann’s Cumulative Hierarchy
We define the cumulative hierarchy.
V0 = ∅,
(Successor case) Vα+1 = P(Vα), for all α, where P(X) is thepowerset of X ,
(Limit case) Vα = ⋃{Vξ ∶ ξ < α}, for all limit α,
V = ⋃{Vα ∶ α ∈ ORD}.
The theory ZF formalizes the first order theory of V.
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Von Neumann’s Cumulative Hierarchy
We define the cumulative hierarchy.
V0 = ∅,
(Successor case) Vα+1 = P(Vα), for all α, where P(X) is thepowerset of X ,
(Limit case) Vα = ⋃{Vξ ∶ ξ < α}, for all limit α,
V = ⋃{Vα ∶ α ∈ ORD}.
The theory ZF formalizes the first order theory of V.
![Page 30: Introduction to Forcing](https://reader030.vdocument.in/reader030/viewer/2022012614/619c174445dca568d06c166b/html5/thumbnails/30.jpg)
Von Neumann’s Cumulative Hierarchy
We define the cumulative hierarchy.
V0 = ∅,
(Successor case) Vα+1 = P(Vα), for all α, where P(X) is thepowerset of X ,
(Limit case) Vα = ⋃{Vξ ∶ ξ < α}, for all limit α,
V = ⋃{Vα ∶ α ∈ ORD}.
The theory ZF formalizes the first order theory of V.
![Page 31: Introduction to Forcing](https://reader030.vdocument.in/reader030/viewer/2022012614/619c174445dca568d06c166b/html5/thumbnails/31.jpg)
Von Neumann’s Cumulative Hierarchy
We define the cumulative hierarchy.
V0 = ∅,
(Successor case) Vα+1 = P(Vα), for all α, where P(X) is thepowerset of X ,
(Limit case) Vα = ⋃{Vξ ∶ ξ < α}, for all limit α,
V = ⋃{Vα ∶ α ∈ ORD}.
The theory ZF formalizes the first order theory of V.
![Page 32: Introduction to Forcing](https://reader030.vdocument.in/reader030/viewer/2022012614/619c174445dca568d06c166b/html5/thumbnails/32.jpg)
Von Neumann’s Cumulative Hierarchy
We define the cumulative hierarchy.
V0 = ∅,
(Successor case) Vα+1 = P(Vα), for all α, where P(X) is thepowerset of X ,
(Limit case) Vα = ⋃{Vξ ∶ ξ < α}, for all limit α,
V = ⋃{Vα ∶ α ∈ ORD}.
The theory ZF formalizes the first order theory of V.
![Page 33: Introduction to Forcing](https://reader030.vdocument.in/reader030/viewer/2022012614/619c174445dca568d06c166b/html5/thumbnails/33.jpg)
Von Neumann’s Cumulative Hierarchy
We define the cumulative hierarchy.
V0 = ∅,
(Successor case) Vα+1 = P(Vα), for all α, where P(X) is thepowerset of X ,
(Limit case) Vα = ⋃{Vξ ∶ ξ < α}, for all limit α,
V = ⋃{Vα ∶ α ∈ ORD}.
The theory ZF formalizes the first order theory of V.
![Page 34: Introduction to Forcing](https://reader030.vdocument.in/reader030/viewer/2022012614/619c174445dca568d06c166b/html5/thumbnails/34.jpg)
Von Neumann’s Cumulative Hierarchy
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What about Choice?
And what about the Axiom of Choice? Well, it is necessary for somebasic theorems in mathematics...
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What about Choice?
On the other hand it leads to some strange paradoxes...
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What about Choice?
And in some countries it is still the topic of hot debate...
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Outline
1 A brief history of Set Theory
2 Independence results
3 ForcingGeneralitiesFundamental theorem of forcingExamples
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Models of ZF
ZF is a first order theory, so we can consider models of ZF. A modelof ZF is a set M with a binary relation E such that (M,E) ⊧ ZF.Note that E may not be the true membership relation ∈.
ZF is recursive and contains arithmetic, hence by Godel’sIncompleteness theorem, if it is consistent then it is incomplete. Infact, ZF does not prove its own consistency.
But, wait! Isn’t V a model of ZF?
Yes! But V is a proper class and the statement that V is a model ofZF cannot even be expressed as a first order statement by Tarski’sundefinability of truth.
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Models of ZF
ZF is a first order theory, so we can consider models of ZF. A modelof ZF is a set M with a binary relation E such that (M,E) ⊧ ZF.Note that E may not be the true membership relation ∈.
ZF is recursive and contains arithmetic, hence by Godel’sIncompleteness theorem, if it is consistent then it is incomplete. Infact, ZF does not prove its own consistency.
But, wait! Isn’t V a model of ZF?
Yes! But V is a proper class and the statement that V is a model ofZF cannot even be expressed as a first order statement by Tarski’sundefinability of truth.
![Page 41: Introduction to Forcing](https://reader030.vdocument.in/reader030/viewer/2022012614/619c174445dca568d06c166b/html5/thumbnails/41.jpg)
Models of ZF
ZF is a first order theory, so we can consider models of ZF. A modelof ZF is a set M with a binary relation E such that (M,E) ⊧ ZF.Note that E may not be the true membership relation ∈.
ZF is recursive and contains arithmetic, hence by Godel’sIncompleteness theorem, if it is consistent then it is incomplete. Infact, ZF does not prove its own consistency.
But, wait! Isn’t V a model of ZF?
Yes! But V is a proper class and the statement that V is a model ofZF cannot even be expressed as a first order statement by Tarski’sundefinability of truth.
![Page 42: Introduction to Forcing](https://reader030.vdocument.in/reader030/viewer/2022012614/619c174445dca568d06c166b/html5/thumbnails/42.jpg)
Models of ZF
ZF is a first order theory, so we can consider models of ZF. A modelof ZF is a set M with a binary relation E such that (M,E) ⊧ ZF.Note that E may not be the true membership relation ∈.
ZF is recursive and contains arithmetic, hence by Godel’sIncompleteness theorem, if it is consistent then it is incomplete. Infact, ZF does not prove its own consistency.
But, wait! Isn’t V a model of ZF?
Yes! But V is a proper class and the statement that V is a model ofZF cannot even be expressed as a first order statement by Tarski’sundefinability of truth.
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Relative consistency of CH and AC
Theorem (Kurt Godel, 1940)If the theory ZF is consistent, then so is ZFC +CH.
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Relative consistency of CH and AC
Theorem (Kurt Godel, 1940)If the theory ZF is consistent, then so is ZFC +CH.
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Effective cumulative hierarchy: L
The definable power setFor each set X , PDef(X) denotes the set of all Y ⊆X which arelogically definable in the structure (X, ∈).
● (AC) PDef(X) = P(X) if and only if X is finite.
Godel’s constructible universe, LDefine Lα by induction on α as follows.
1 L0 = ∅,2 (Successor case) Lα+1 = PDef(Lα),3 (Limit case) Lα = ⋃{Lβ ∶ β < α}, if α is limit,4 L = ⋃{Lα ∶ α ∈ ORD}.
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Effective cumulative hierarchy: L
The definable power setFor each set X , PDef(X) denotes the set of all Y ⊆X which arelogically definable in the structure (X, ∈).
● (AC) PDef(X) = P(X) if and only if X is finite.
Godel’s constructible universe, LDefine Lα by induction on α as follows.
1 L0 = ∅,2 (Successor case) Lα+1 = PDef(Lα),3 (Limit case) Lα = ⋃{Lβ ∶ β < α}, if α is limit,4 L = ⋃{Lα ∶ α ∈ ORD}.
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Effective cumulative hierarchy: L
The definable power setFor each set X , PDef(X) denotes the set of all Y ⊆X which arelogically definable in the structure (X, ∈).
● (AC) PDef(X) = P(X) if and only if X is finite.
Godel’s constructible universe, LDefine Lα by induction on α as follows.
1 L0 = ∅,2 (Successor case) Lα+1 = PDef(Lα),3 (Limit case) Lα = ⋃{Lβ ∶ β < α}, if α is limit,4 L = ⋃{Lα ∶ α ∈ ORD}.
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L is a proper class, i.e. not a set. Formally, we prove the followingmetatheorem.
Theorem1 ZF ⊢ (ϕ)L, for every axiom ϕ of ZF,2 ZF ⊢ (V = L)L,3 ZF ⊢ V = L→ CH, and4 ZF ⊢ V = L→ AC.
L is the smallest transitive class which is a model of ZF, hence withthis method we cannot prove the independence of CH and AC.
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L is a proper class, i.e. not a set. Formally, we prove the followingmetatheorem.
Theorem1 ZF ⊢ (ϕ)L, for every axiom ϕ of ZF,2 ZF ⊢ (V = L)L,3 ZF ⊢ V = L→ CH, and4 ZF ⊢ V = L→ AC.
L is the smallest transitive class which is a model of ZF, hence withthis method we cannot prove the independence of CH and AC.
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Independence of CH and AC
Theorem (Paul Cohen, 1963)If the theory ZF is consistent, then so are the theories ZFC + ¬CHand ZF + ¬AC.
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Independence of CH and AC
Theorem (Paul Cohen, 1963)If the theory ZF is consistent, then so are the theories ZFC + ¬CHand ZF + ¬AC.
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Outline
1 A brief history of Set Theory
2 Independence results
3 ForcingGeneralitiesFundamental theorem of forcingExamples
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Cohen’s method of forcing
There are two equivalent ways of presenting forcing.
One is to work in V , but change the concept of truth. We fix acomplete Boolean algebra B and define the B-valued universe VB. Ifϕ(x1, . . . , xn) is a formula of set theory, and τ1, . . . , τn ∈ VB, we candefine ∣∣ϕ(τ1, . . . , τn)∣∣, the B-value of ϕ, which measure how muchϕ(τ1, . . . , τn) is true in VB. Then we show that ∣∣ϕ∣∣ = 1B, for everyaxiom ϕ of ZF. Moreover, if ϕ1, . . . , ϕn ⊢ ψ then
∣∣ϕ1∣∣ ∧ . . . ∧ ∣∣ϕn∣∣ ≤ ∣∣ψ∣∣.
Then, by choosing carefully B, we can make ∣∣CH∣∣ equal to 0B or 1B.
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Cohen’s method of forcing
There are two equivalent ways of presenting forcing.
One is to work in V , but change the concept of truth. We fix acomplete Boolean algebra B and define the B-valued universe VB. Ifϕ(x1, . . . , xn) is a formula of set theory, and τ1, . . . , τn ∈ VB, we candefine ∣∣ϕ(τ1, . . . , τn)∣∣, the B-value of ϕ, which measure how muchϕ(τ1, . . . , τn) is true in VB. Then we show that ∣∣ϕ∣∣ = 1B, for everyaxiom ϕ of ZF. Moreover, if ϕ1, . . . , ϕn ⊢ ψ then
∣∣ϕ1∣∣ ∧ . . . ∧ ∣∣ϕn∣∣ ≤ ∣∣ψ∣∣.
Then, by choosing carefully B, we can make ∣∣CH∣∣ equal to 0B or 1B.
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Cohen’s method of forcing
There are two equivalent ways of presenting forcing.
One is to work in V , but change the concept of truth. We fix acomplete Boolean algebra B and define the B-valued universe VB. Ifϕ(x1, . . . , xn) is a formula of set theory, and τ1, . . . , τn ∈ VB, we candefine ∣∣ϕ(τ1, . . . , τn)∣∣, the B-value of ϕ, which measure how muchϕ(τ1, . . . , τn) is true in VB. Then we show that ∣∣ϕ∣∣ = 1B, for everyaxiom ϕ of ZF. Moreover, if ϕ1, . . . , ϕn ⊢ ψ then
∣∣ϕ1∣∣ ∧ . . . ∧ ∣∣ϕn∣∣ ≤ ∣∣ψ∣∣.
Then, by choosing carefully B, we can make ∣∣CH∣∣ equal to 0B or 1B.
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Cohen’s method of forcing
The second method is to assume that there is a countable, transitiveset M such that (M, ∈) satisfies ZFC and work with actual models.Given a formula ϕ the truth value of ϕ may change when we changethe model, so we must be careful.
∆0-formulasA formula ϕ of set theory is ∆0 if every quantifier ϕ is bounded, i.e.is of the form ∃x ∈ y or ∀x ∈ y, for some variables x and y.
Absoluteness of ∆0-formulasIf M is a transitive set, ϕ(v) a ∆0-formula and a ∈M . ThenM ⊧ ϕ(a) iff V ⊧ ϕ(a).
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Cohen’s method of forcing
The second method is to assume that there is a countable, transitiveset M such that (M, ∈) satisfies ZFC and work with actual models.Given a formula ϕ the truth value of ϕ may change when we changethe model, so we must be careful.
∆0-formulasA formula ϕ of set theory is ∆0 if every quantifier ϕ is bounded, i.e.is of the form ∃x ∈ y or ∀x ∈ y, for some variables x and y.
Absoluteness of ∆0-formulasIf M is a transitive set, ϕ(v) a ∆0-formula and a ∈M . ThenM ⊧ ϕ(a) iff V ⊧ ϕ(a).
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Cohen’s method of forcing
The second method is to assume that there is a countable, transitiveset M such that (M, ∈) satisfies ZFC and work with actual models.Given a formula ϕ the truth value of ϕ may change when we changethe model, so we must be careful.
∆0-formulasA formula ϕ of set theory is ∆0 if every quantifier ϕ is bounded, i.e.is of the form ∃x ∈ y or ∀x ∈ y, for some variables x and y.
Absoluteness of ∆0-formulasIf M is a transitive set, ϕ(v) a ∆0-formula and a ∈M . ThenM ⊧ ϕ(a) iff V ⊧ ϕ(a).
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Cohen’s method of forcing
So, fix our ctm M and work for a while in M .
Forcing notionsA forcing notion is a partial order (P,≤) with the largest element 1P .
ConditionsElements of P are called conditions. If p ≤ q we say that p isstronger than q. If there is r such that r ≤ p, q we say that p and q arecompatible. Otherwise, we say that they are incompatible and wewrite p ⊥ q. A set of incompatible conditions is called an antichain.
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Cohen’s method of forcing
So, fix our ctm M and work for a while in M .
Forcing notionsA forcing notion is a partial order (P,≤) with the largest element 1P .
ConditionsElements of P are called conditions. If p ≤ q we say that p isstronger than q. If there is r such that r ≤ p, q we say that p and q arecompatible. Otherwise, we say that they are incompatible and wewrite p ⊥ q. A set of incompatible conditions is called an antichain.
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Cohen’s method of forcing
So, fix our ctm M and work for a while in M .
Forcing notionsA forcing notion is a partial order (P,≤) with the largest element 1P .
ConditionsElements of P are called conditions. If p ≤ q we say that p isstronger than q. If there is r such that r ≤ p, q we say that p and q arecompatible. Otherwise, we say that they are incompatible and wewrite p ⊥ q. A set of incompatible conditions is called an antichain.
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Cohen’s method of forcing
Dense setsD ⊆ P is called dense if for every p ∈ P there is q ∈D with q ≤ p.
FiltersA subset F of P is called a filter if:
1 if p, q ∈ F then there is r ≤ p, q with r ∈ F ,2 if p ∈ F and p ≤ q then q ∈ F .
Generic filtersA filter G is M -generic if G ∩D ≠ ∅, for all dense D ⊆ P withD ∈M .
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Cohen’s method of forcing
Dense setsD ⊆ P is called dense if for every p ∈ P there is q ∈D with q ≤ p.
FiltersA subset F of P is called a filter if:
1 if p, q ∈ F then there is r ≤ p, q with r ∈ F ,2 if p ∈ F and p ≤ q then q ∈ F .
Generic filtersA filter G is M -generic if G ∩D ≠ ∅, for all dense D ⊆ P withD ∈M .
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Cohen’s method of forcing
Dense setsD ⊆ P is called dense if for every p ∈ P there is q ∈D with q ≤ p.
FiltersA subset F of P is called a filter if:
1 if p, q ∈ F then there is r ≤ p, q with r ∈ F ,2 if p ∈ F and p ≤ q then q ∈ F .
Generic filtersA filter G is M -generic if G ∩D ≠ ∅, for all dense D ⊆ P withD ∈M .
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Cohen’s method of forcing
In nontrivial cases there are no M -generic filters in M , but it is easyto construct them in V.
Baire category theoremIn V, for every p ∈ P , there is an M -generic filter G such that p ∈ G.
Proof.M is countable, so we can list all dense subsets of P which belong toM as D0,D1, . . .. Then we build a sequence p0 ≥ p1 ≥ . . .. Let p0 = p.Given pn, use the fact that Dn is dense to pick pn+1 ∈Dn such thatpn+1 ≤ pn. Finally, let G = {q ∈ P ∶ ∃npn ≤ q}.
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Cohen’s method of forcing
In nontrivial cases there are no M -generic filters in M , but it is easyto construct them in V.
Baire category theoremIn V, for every p ∈ P , there is an M -generic filter G such that p ∈ G.
Proof.M is countable, so we can list all dense subsets of P which belong toM as D0,D1, . . .. Then we build a sequence p0 ≥ p1 ≥ . . .. Let p0 = p.Given pn, use the fact that Dn is dense to pick pn+1 ∈Dn such thatpn+1 ≤ pn. Finally, let G = {q ∈ P ∶ ∃npn ≤ q}.
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Cohen’s method of forcing
In nontrivial cases there are no M -generic filters in M , but it is easyto construct them in V.
Baire category theoremIn V, for every p ∈ P , there is an M -generic filter G such that p ∈ G.
Proof.M is countable, so we can list all dense subsets of P which belong toM as D0,D1, . . .. Then we build a sequence p0 ≥ p1 ≥ . . .. Let p0 = p.Given pn, use the fact that Dn is dense to pick pn+1 ∈Dn such thatpn+1 ≤ pn. Finally, let G = {q ∈ P ∶ ∃npn ≤ q}.
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Example of a forcing notion
DefinitionLet P0 consist of all finite partial functions from ω to {0,1}. Theorder is given by: p ≤ q iff q ⊆ p.
This definition is done in M , but it gives the same object in V. Whatcan we say about an M -generic filter G?
1 If p, q ∈ G then p ∪ q is a function.2 Let g = ⋃G. Then g is a total function from ω to {0,1}.3 Let xg = {n ∶ g(n) = 1}. Then xg is infinite and co-infinite.4 xg ∉M .
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Example of a forcing notion
DefinitionLet P0 consist of all finite partial functions from ω to {0,1}. Theorder is given by: p ≤ q iff q ⊆ p.
This definition is done in M , but it gives the same object in V. Whatcan we say about an M -generic filter G?
1 If p, q ∈ G then p ∪ q is a function.2 Let g = ⋃G. Then g is a total function from ω to {0,1}.3 Let xg = {n ∶ g(n) = 1}. Then xg is infinite and co-infinite.4 xg ∉M .
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1 G is a filter, so if p, q ∈ G there is r ≤ p, q. Hence p ∪ q is afunction.
2 Given n, let Dn = {p ∈ P0 ∶ n ∈ dom(p)}. Then Dn is dense andG ∩Dn ≠ ∅, so n ∈ dom(g).
3 E0n = {p ∈ P0 ∶ ∣p
−1(0)∣ ≥ n} and E1n = {p ∈ P0 ∶ ∣p
−1(1)∣ ≥ n}.Then E0
n and E1n are dense, for all n, and hence intersect G.
4 Given a real z ∈M (think of z ∶ ω → {0,1}), let
Hz = {p ∈ P0 ∶ ∃n ∈ dom(p)p(n) ≠ z(n)}.
Then Hz is dense and intersects G, for all z ∈M .
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1 G is a filter, so if p, q ∈ G there is r ≤ p, q. Hence p ∪ q is afunction.
2 Given n, let Dn = {p ∈ P0 ∶ n ∈ dom(p)}. Then Dn is dense andG ∩Dn ≠ ∅, so n ∈ dom(g).
3 E0n = {p ∈ P0 ∶ ∣p
−1(0)∣ ≥ n} and E1n = {p ∈ P0 ∶ ∣p
−1(1)∣ ≥ n}.Then E0
n and E1n are dense, for all n, and hence intersect G.
4 Given a real z ∈M (think of z ∶ ω → {0,1}), let
Hz = {p ∈ P0 ∶ ∃n ∈ dom(p)p(n) ≠ z(n)}.
Then Hz is dense and intersects G, for all z ∈M .
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1 G is a filter, so if p, q ∈ G there is r ≤ p, q. Hence p ∪ q is afunction.
2 Given n, let Dn = {p ∈ P0 ∶ n ∈ dom(p)}. Then Dn is dense andG ∩Dn ≠ ∅, so n ∈ dom(g).
3 E0n = {p ∈ P0 ∶ ∣p
−1(0)∣ ≥ n} and E1n = {p ∈ P0 ∶ ∣p
−1(1)∣ ≥ n}.Then E0
n and E1n are dense, for all n, and hence intersect G.
4 Given a real z ∈M (think of z ∶ ω → {0,1}), let
Hz = {p ∈ P0 ∶ ∃n ∈ dom(p)p(n) ≠ z(n)}.
Then Hz is dense and intersects G, for all z ∈M .
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1 G is a filter, so if p, q ∈ G there is r ≤ p, q. Hence p ∪ q is afunction.
2 Given n, let Dn = {p ∈ P0 ∶ n ∈ dom(p)}. Then Dn is dense andG ∩Dn ≠ ∅, so n ∈ dom(g).
3 E0n = {p ∈ P0 ∶ ∣p
−1(0)∣ ≥ n} and E1n = {p ∈ P0 ∶ ∣p
−1(1)∣ ≥ n}.Then E0
n and E1n are dense, for all n, and hence intersect G.
4 Given a real z ∈M (think of z ∶ ω → {0,1}), let
Hz = {p ∈ P0 ∶ ∃n ∈ dom(p)p(n) ≠ z(n)}.
Then Hz is dense and intersects G, for all z ∈M .
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Fundamental theorem of forcing I
The fundamental theorem of forcing ILet M be a ctm of ZFC, (P,≤) ∈M a forcing notion and G anM -generic filter. Then there is a transitive set M[G] such that:
1 M ∪ {G} ⊆M[G],2 M[G] ∩ORD = M ∩ORD,3 M[G] ⊧ ZFC,4 M[G] is minimal with the above properties.
M[G] is obtained by adding G to M and closing under simpleset-theoretic operations.
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Fundamental theorem of forcing I
The fundamental theorem of forcing ILet M be a ctm of ZFC, (P,≤) ∈M a forcing notion and G anM -generic filter. Then there is a transitive set M[G] such that:
1 M ∪ {G} ⊆M[G],2 M[G] ∩ORD = M ∩ORD,3 M[G] ⊧ ZFC,4 M[G] is minimal with the above properties.
M[G] is obtained by adding G to M and closing under simpleset-theoretic operations.
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P -names
People living in M do not know G but they can still talk about M[G].Every t ∈M[G] will have a name τ ∈M . In general, τ is not unique.One can interpret τ only once G is known. The following definition isdone in M by ∈∗-induction.
P -names∅ is a P -name. We say that τ is a P -name if every element of τ is ofthe form (q, σ), where q ∈ P and σ is a P -name. Let MP be the(class) of all P -names.
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P -names
People living in M do not know G but they can still talk about M[G].Every t ∈M[G] will have a name τ ∈M . In general, τ is not unique.One can interpret τ only once G is known. The following definition isdone in M by ∈∗-induction.
P -names∅ is a P -name. We say that τ is a P -name if every element of τ is ofthe form (q, σ), where q ∈ P and σ is a P -name. Let MP be the(class) of all P -names.
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Fundamental theorem of forcing II
Let G be an M -generic filter. We define KG(τ) for every P -name τ .1 KG(∅) = ∅,2 KG(τ) = {KG(σ) ∶ ∃q ∈ G (q, σ) ∈ τ}.
The fundamental theorem of forcing IIM[G] = {KG(τ) ∶ τ ∈M and τ is a P -name}.
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Fundamental theorem of forcing II
Let G be an M -generic filter. We define KG(τ) for every P -name τ .1 KG(∅) = ∅,2 KG(τ) = {KG(σ) ∶ ∃q ∈ G (q, σ) ∈ τ}.
The fundamental theorem of forcing IIM[G] = {KG(τ) ∶ τ ∈M and τ is a P -name}.
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Canonical names
How do we show that M ⊆M[G] and G ∈M[G]? First, we build aname for every element of M .
Canonical namesLet ∅ = ∅. If x ≠ ∅ let x = {(1P , y) ∶ y ∈ x}.
Since 1P ∈ G, it is easy to check that KG(x) = x, for all x ∈M .
DefinitionLet Γ = {(p, p) ∶ p ∈ P}.
Then KG(Γ) = G, i.e. every generic filter G interprets Γ as itself!
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Canonical names
How do we show that M ⊆M[G] and G ∈M[G]? First, we build aname for every element of M .
Canonical namesLet ∅ = ∅. If x ≠ ∅ let x = {(1P , y) ∶ y ∈ x}.
Since 1P ∈ G, it is easy to check that KG(x) = x, for all x ∈M .
DefinitionLet Γ = {(p, p) ∶ p ∈ P}.
Then KG(Γ) = G, i.e. every generic filter G interprets Γ as itself!
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Canonical names
How do we show that M ⊆M[G] and G ∈M[G]? First, we build aname for every element of M .
Canonical namesLet ∅ = ∅. If x ≠ ∅ let x = {(1P , y) ∶ y ∈ x}.
Since 1P ∈ G, it is easy to check that KG(x) = x, for all x ∈M .
DefinitionLet Γ = {(p, p) ∶ p ∈ P}.
Then KG(Γ) = G, i.e. every generic filter G interprets Γ as itself!
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Language of forcing
The language Lf of forcing consists of symbols ∈, =, a unarypredicate S and a constant τ , for every P -name τ . We interpret Lf inM[G]. We let ∈ and = be as usual, τ is interpreted by KG(τ), forevery P -name τ . Finally, we let S(x) iff x ∈M .
Forcing relationLet p ∈ P , ϕ a formula of Lf , and τ1, . . . , τn the P -names appearingin ϕ. We say that p forces ϕ and write p ⊩ ϕ iff, for every M -genericfilter G with p ∈ G, we have
M[G] ⊧ ϕ(KG(τ1), . . . ,KG(τn)).
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Language of forcing
The language Lf of forcing consists of symbols ∈, =, a unarypredicate S and a constant τ , for every P -name τ . We interpret Lf inM[G]. We let ∈ and = be as usual, τ is interpreted by KG(τ), forevery P -name τ . Finally, we let S(x) iff x ∈M .
Forcing relationLet p ∈ P , ϕ a formula of Lf , and τ1, . . . , τn the P -names appearingin ϕ. We say that p forces ϕ and write p ⊩ ϕ iff, for every M -genericfilter G with p ∈ G, we have
M[G] ⊧ ϕ(KG(τ1), . . . ,KG(τn)).
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Fundamental theorem of forcing III and IV
The fundamental theorem of forcing IIILet ϕ be a closed formula of Lf and G an M -generic filter. Then
M[G] ⊧ ϕ if and only if p ⊩ ϕ, for some p ∈ G.
The fundamental theorem of forcing IV - definability of theforcing relationIf ϕ(x1, . . . , xn) is a formula of LZF ∪ {S}, then there is a formulaθ(y, z, x1, . . . , xn) such that, for every forcing notion (P,≤), p ∈ P ,and P -names τ1, . . . , τn
p ⊩ ϕ(τ1, . . . , τn) iff θ(P, p, τ1, . . . , τn).
At first sight, this looks surprising. M does not have any genericfilter, yet somehow it is able to talk about all generic filters.
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Fundamental theorem of forcing III and IV
The fundamental theorem of forcing IIILet ϕ be a closed formula of Lf and G an M -generic filter. Then
M[G] ⊧ ϕ if and only if p ⊩ ϕ, for some p ∈ G.
The fundamental theorem of forcing IV - definability of theforcing relationIf ϕ(x1, . . . , xn) is a formula of LZF ∪ {S}, then there is a formulaθ(y, z, x1, . . . , xn) such that, for every forcing notion (P,≤), p ∈ P ,and P -names τ1, . . . , τn
p ⊩ ϕ(τ1, . . . , τn) iff θ(P, p, τ1, . . . , τn).
At first sight, this looks surprising. M does not have any genericfilter, yet somehow it is able to talk about all generic filters.
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Fundamental theorem of forcing III and IV
The fundamental theorem of forcing IIILet ϕ be a closed formula of Lf and G an M -generic filter. Then
M[G] ⊧ ϕ if and only if p ⊩ ϕ, for some p ∈ G.
The fundamental theorem of forcing IV - definability of theforcing relationIf ϕ(x1, . . . , xn) is a formula of LZF ∪ {S}, then there is a formulaθ(y, z, x1, . . . , xn) such that, for every forcing notion (P,≤), p ∈ P ,and P -names τ1, . . . , τn
p ⊩ ϕ(τ1, . . . , τn) iff θ(P, p, τ1, . . . , τn).
At first sight, this looks surprising. M does not have any genericfilter, yet somehow it is able to talk about all generic filters.
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Forcing relation - atomic case
In M , we define p ⊩ τ1 = τ2, p ⊩ τ1 ⊆ τ2 and p ⊩ τ1 ∈ τ2, for p ∈ Pand P -names τ1, τ2 by induction on (rank(τ1), rank(τ2)).
Definition1 p ⊩ τ1 = τ2 iff p ⊩ τ1 ⊆ τ2 and p ⊩ τ2 ⊆ τ1.2 p ⊩ τ1 ⊆ τ2 iff for every (q, σ) ∈ τ1 and r ≤ p, q there is s ≤ r
such that s ⊩ σ ∈ τ2.3 p ⊩ τ1 ∈ τ2 iff for every q ≤ p there is (r, σ) ∈ τ2 and s ≤ q, r such
that s ⊩ τ1 = σ.
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Forcing relation - atomic case
In M , we define p ⊩ τ1 = τ2, p ⊩ τ1 ⊆ τ2 and p ⊩ τ1 ∈ τ2, for p ∈ Pand P -names τ1, τ2 by induction on (rank(τ1), rank(τ2)).
Definition1 p ⊩ τ1 = τ2 iff p ⊩ τ1 ⊆ τ2 and p ⊩ τ2 ⊆ τ1.2 p ⊩ τ1 ⊆ τ2 iff for every (q, σ) ∈ τ1 and r ≤ p, q there is s ≤ r
such that s ⊩ σ ∈ τ2.3 p ⊩ τ1 ∈ τ2 iff for every q ≤ p there is (r, σ) ∈ τ2 and s ≤ q, r such
that s ⊩ τ1 = σ.
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Forcing relation - connectives and quantifiers
Still in M , we continue to define p ⊩ ϕ, for non atomic ϕ.
Definition1 p ⊩ ϕ ∧ ψ iff p ⊩ ϕ and p ⊩ ψ.2 p ⊩ ¬ϕ iff q ⊮ ϕ, for all q ≤ p.3 p ⊩ ∃xϕ(x) iff for all q ≤ p there is r ≤ q and a P -name τ such
that r ⊩ ϕ(τ).
Proposition1 If p ⊩ ϕ and q ≤ p then q ⊩ ϕ.2 {p ∶ p ⊩ ϕ or p ⊩ ¬ϕ} is dense.3 No p forces both ϕ and ¬ϕ.
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Forcing relation - connectives and quantifiers
Still in M , we continue to define p ⊩ ϕ, for non atomic ϕ.
Definition1 p ⊩ ϕ ∧ ψ iff p ⊩ ϕ and p ⊩ ψ.2 p ⊩ ¬ϕ iff q ⊮ ϕ, for all q ≤ p.3 p ⊩ ∃xϕ(x) iff for all q ≤ p there is r ≤ q and a P -name τ such
that r ⊩ ϕ(τ).
Proposition1 If p ⊩ ϕ and q ≤ p then q ⊩ ϕ.2 {p ∶ p ⊩ ϕ or p ⊩ ¬ϕ} is dense.3 No p forces both ϕ and ¬ϕ.
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Forcing relation - connectives and quantifiers
Still in M , we continue to define p ⊩ ϕ, for non atomic ϕ.
Definition1 p ⊩ ϕ ∧ ψ iff p ⊩ ϕ and p ⊩ ψ.2 p ⊩ ¬ϕ iff q ⊮ ϕ, for all q ≤ p.3 p ⊩ ∃xϕ(x) iff for all q ≤ p there is r ≤ q and a P -name τ such
that r ⊩ ϕ(τ).
Proposition1 If p ⊩ ϕ and q ≤ p then q ⊩ ϕ.2 {p ∶ p ⊩ ϕ or p ⊩ ¬ϕ} is dense.3 No p forces both ϕ and ¬ϕ.
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Proof of the Fundamental theorem of forcing
The proof of the Fundamental theorem of forcing is a straightforward,but tedious exercise. We prove:
M[G] ⊧ ϕ(KG(τ1), . . . ,KG(τn)) iff ∃p ∈ Gp ⊩ ϕ(τ1, . . . , τn).
1 First the atomic case - requires careful transfinite induction2 Then the connectives and quantifier case - easy.
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Proof of the Fundamental theorem of forcing
The proof of the Fundamental theorem of forcing is a straightforward,but tedious exercise. We prove:
M[G] ⊧ ϕ(KG(τ1), . . . ,KG(τn)) iff ∃p ∈ Gp ⊩ ϕ(τ1, . . . , τn).
1 First the atomic case - requires careful transfinite induction2 Then the connectives and quantifier case - easy.
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Proof of the Fundamental theorem of forcing
The proof of the Fundamental theorem of forcing is a straightforward,but tedious exercise. We prove:
M[G] ⊧ ϕ(KG(τ1), . . . ,KG(τn)) iff ∃p ∈ Gp ⊩ ϕ(τ1, . . . , τn).
1 First the atomic case - requires careful transfinite induction2 Then the connectives and quantifier case - easy.
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Proof of the Fundamental theorem of forcing
LemmaM[G] ⊧ ZFC.
Proof.1 Extensionality: M[G] is transitive2 Foundation: holds in each ∈ model3 For those axioms that asserts the existence of sets, we need to
design appropriate names.
LemmaIf N is a transitive model of ZF such that M ⊆ N and G ∈ N thenM[G] ⊆ N .
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Proof of the Fundamental theorem of forcing
LemmaM[G] ⊧ ZFC.
Proof.1 Extensionality: M[G] is transitive2 Foundation: holds in each ∈ model3 For those axioms that asserts the existence of sets, we need to
design appropriate names.
LemmaIf N is a transitive model of ZF such that M ⊆ N and G ∈ N thenM[G] ⊆ N .
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Proof of the Fundamental theorem of forcing
LemmaM[G] ⊧ ZFC.
Proof.1 Extensionality: M[G] is transitive2 Foundation: holds in each ∈ model3 For those axioms that asserts the existence of sets, we need to
design appropriate names.
LemmaIf N is a transitive model of ZF such that M ⊆ N and G ∈ N thenM[G] ⊆ N .
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Finite partial functions
In applications, the hard part is designing the forcing notion that doeswhat we want. We give a simple example.
Finite partial functionsGiven sets I, J let Fn(I, J) consist of all finite partial functions fromI to J . We say: p ≤ q iff q ⊆ p.
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Finite partial functions
In applications, the hard part is designing the forcing notion that doeswhat we want. We give a simple example.
Finite partial functionsGiven sets I, J let Fn(I, J) consist of all finite partial functions fromI to J . We say: p ≤ q iff q ⊆ p.
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Finite partial functions
● Collapsing cardinals Let κ > ω be a cardinal in M . Force withFn(ω,κ). Then ⋃G is a total function from ω onto κ. So, κ is not acardinal in M .
● Adding many reals Let κ > ω1 be a cardinal in M . Force withFn(κ × ω,2). Let G be generic. Then:
1 g = ⋃G is a total function from κ × ω → 2.2 For α < κ let gα(n) = g(α,n). Then the the gα are distinct.
So, we made 2ω ≥ κ. But how do we know that κ is not collapsed?
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Finite partial functions
● Collapsing cardinals Let κ > ω be a cardinal in M . Force withFn(ω,κ). Then ⋃G is a total function from ω onto κ. So, κ is not acardinal in M .
● Adding many reals Let κ > ω1 be a cardinal in M . Force withFn(κ × ω,2). Let G be generic. Then:
1 g = ⋃G is a total function from κ × ω → 2.2 For α < κ let gα(n) = g(α,n). Then the the gα are distinct.
So, we made 2ω ≥ κ. But how do we know that κ is not collapsed?
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Finite partial functions
● Collapsing cardinals Let κ > ω be a cardinal in M . Force withFn(ω,κ). Then ⋃G is a total function from ω onto κ. So, κ is not acardinal in M .
● Adding many reals Let κ > ω1 be a cardinal in M . Force withFn(κ × ω,2). Let G be generic. Then:
1 g = ⋃G is a total function from κ × ω → 2.2 For α < κ let gα(n) = g(α,n). Then the the gα are distinct.
So, we made 2ω ≥ κ. But how do we know that κ is not collapsed?
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Countable antichain condition
DefinitionP satisfies the countable antichain condition (c.a.c.) if any antichainA in P is at most countable.
TheoremSuppose P ∈M and M ⊧ ”P satisfies the c.a.c.”. Then, for α ∈M
M[G] ⊧ α is a cardinal iff M ⊧ α is a cardinal.
LemmaFn(κ × ω,2) satisfies the c.a.c.
So, starting from a model M of CH we can make 2ω as large as welike!
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Countable antichain condition
DefinitionP satisfies the countable antichain condition (c.a.c.) if any antichainA in P is at most countable.
TheoremSuppose P ∈M and M ⊧ ”P satisfies the c.a.c.”. Then, for α ∈M
M[G] ⊧ α is a cardinal iff M ⊧ α is a cardinal.
LemmaFn(κ × ω,2) satisfies the c.a.c.
So, starting from a model M of CH we can make 2ω as large as welike!
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Countable antichain condition
DefinitionP satisfies the countable antichain condition (c.a.c.) if any antichainA in P is at most countable.
TheoremSuppose P ∈M and M ⊧ ”P satisfies the c.a.c.”. Then, for α ∈M
M[G] ⊧ α is a cardinal iff M ⊧ α is a cardinal.
LemmaFn(κ × ω,2) satisfies the c.a.c.
So, starting from a model M of CH we can make 2ω as large as welike!
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Countable partial functions
What if we start from M which satisfies ¬CH and want to make CHtrue in M[G]? Easy! All we need to do is collapse 2ω to ω1.
Countable partial functionsGiven I and J , let CPF(I, J) set of countable partial functions fromI to J . Let p ≤ q iff q ⊆ p.
We force with CPF(ω1,2ω) as defined in M . If G is generic then
⋃G is a total function from ωM1 onto (2ω)M .
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Countable partial functions
What if we start from M which satisfies ¬CH and want to make CHtrue in M[G]? Easy! All we need to do is collapse 2ω to ω1.
Countable partial functionsGiven I and J , let CPF(I, J) set of countable partial functions fromI to J . Let p ≤ q iff q ⊆ p.
We force with CPF(ω1,2ω) as defined in M . If G is generic then
⋃G is a total function from ωM1 onto (2ω)M .
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Countable partial functions
What if we start from M which satisfies ¬CH and want to make CHtrue in M[G]? Easy! All we need to do is collapse 2ω to ω1.
Countable partial functionsGiven I and J , let CPF(I, J) set of countable partial functions fromI to J . Let p ≤ q iff q ⊆ p.
We force with CPF(ω1,2ω) as defined in M . If G is generic then
⋃G is a total function from ωM1 onto (2ω)M .
![Page 110: Introduction to Forcing](https://reader030.vdocument.in/reader030/viewer/2022012614/619c174445dca568d06c166b/html5/thumbnails/110.jpg)
Countable partial functions
What if we start from M which satisfies ¬CH and want to make CHtrue in M[G]? Easy! All we need to do is collapse 2ω to ω1.
Countable partial functionsGiven I and J , let CPF(I, J) set of countable partial functions fromI to J . Let p ≤ q iff q ⊆ p.
We force with CPF(ω1,2ω) as defined in M . If G is generic then
⋃G is a total function from ωM1 onto (2ω)M .
![Page 111: Introduction to Forcing](https://reader030.vdocument.in/reader030/viewer/2022012614/619c174445dca568d06c166b/html5/thumbnails/111.jpg)
Countably closed forcing notions
We need to check that ωM[G]1 = ωM1 and that we did not add any reals.
DefinitionP is countably closed if for any decreasing sequence p0 ≥ p1 ≥ . . .there is q such that q ≤ pn, for all n.
Proposition
If P is countably closed then (2ω)M[G] = (2ω)M and ωM[G]1 = ωM1 .
And CPF(ω1,2ω) is countably closed, so M[G] ⊧ CH.
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Countably closed forcing notions
We need to check that ωM[G]1 = ωM1 and that we did not add any reals.
DefinitionP is countably closed if for any decreasing sequence p0 ≥ p1 ≥ . . .there is q such that q ≤ pn, for all n.
Proposition
If P is countably closed then (2ω)M[G] = (2ω)M and ωM[G]1 = ωM1 .
And CPF(ω1,2ω) is countably closed, so M[G] ⊧ CH.
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Countably closed forcing notions
We need to check that ωM[G]1 = ωM1 and that we did not add any reals.
DefinitionP is countably closed if for any decreasing sequence p0 ≥ p1 ≥ . . .there is q such that q ≤ pn, for all n.
Proposition
If P is countably closed then (2ω)M[G] = (2ω)M and ωM[G]1 = ωM1 .
And CPF(ω1,2ω) is countably closed, so M[G] ⊧ CH.
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Negation of the Axiom of Choice
QuestionIf M ⊧ AC then so does M[G]. So, how do we get a model of ¬AC?
SketchStart with M , force with Fn(ω,ω) to get M[G]. Then define anintermediate model N , i.e. M ⊆ N ⊆M[G], such that N ⊧ ¬AC. Nis a symmetric model, i.e. there is a group Σ in M acting on MP andN is the set of all KG(τ), for τ a P -name invariant under all σ ∈ Σ.
Details some other time....
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Negation of the Axiom of Choice
QuestionIf M ⊧ AC then so does M[G]. So, how do we get a model of ¬AC?
SketchStart with M , force with Fn(ω,ω) to get M[G]. Then define anintermediate model N , i.e. M ⊆ N ⊆M[G], such that N ⊧ ¬AC. Nis a symmetric model, i.e. there is a group Σ in M acting on MP andN is the set of all KG(τ), for τ a P -name invariant under all σ ∈ Σ.
Details some other time....
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Negation of the Axiom of Choice
QuestionIf M ⊧ AC then so does M[G]. So, how do we get a model of ¬AC?
SketchStart with M , force with Fn(ω,ω) to get M[G]. Then define anintermediate model N , i.e. M ⊆ N ⊆M[G], such that N ⊧ ¬AC. Nis a symmetric model, i.e. there is a group Σ in M acting on MP andN is the set of all KG(τ), for τ a P -name invariant under all σ ∈ Σ.
Details some other time....
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