Forcing axioms and cardinal arithmetic II

82 Pages·2006·1.21 MB·English

by  Boban Velickovic

Forcing axioms
and
cardinal arithmetic
II
Boban Velickovic
EquipedeLogique
Universite´ deParis7
Logic Colloquium 2006, Nijmegen,
July 27- August 2 2006
TheMappingReflectionPrinciple TheP-idealdichotomy
Outline
1 The Mapping Reflection Principle
The statement of MRP
MRP and the continuum
MRP and (cid:3)(κ)
2 The P-ideal dichotomy
P-ideals
The Singular Cardinal Hypothesis
TheMappingReflectionPrinciple TheP-idealdichotomy
Outline
1 The Mapping Reflection Principle
The statement of MRP
MRP and the continuum
MRP and (cid:3)(κ)
2 The P-ideal dichotomy
P-ideals
The Singular Cardinal Hypothesis
TheMappingReflectionPrinciple TheP-idealdichotomy
ThestatementofMRP
The statement of MRP
Definition
Let θ be a regular cardinal, let X be uncountable, and let
M ≺ H be countable such that [X]ω ∈ M. A subset Σ of [X]ω is
θ
M-stationary iff for all E ∈ M such that E ⊆ [X]ω is club,
Σ∩E ∩M 6= ∅.
If Σ in M then this is just the usual notion, but we will be
interested precisely in the case when Σ is not in M.
Example
LetM ≺ H . Letδ = M ∩ω andsupposeA ⊆ δ hasordertype
ℵ2 1
< δ. Then δ\A is M-stationary.
TheMappingReflectionPrinciple TheP-idealdichotomy
ThestatementofMRP
The statement of MRP
Definition
Let θ be a regular cardinal, let X be uncountable, and let
M ≺ H be countable such that [X]ω ∈ M. A subset Σ of [X]ω is
θ
M-stationary iff for all E ∈ M such that E ⊆ [X]ω is club,
Σ∩E ∩M 6= ∅.
If Σ in M then this is just the usual notion, but we will be
interested precisely in the case when Σ is not in M.
Example
LetM ≺ H . Letδ = M ∩ω andsupposeA ⊆ δ hasordertype
ℵ2 1
< δ. Then δ\A is M-stationary.
TheMappingReflectionPrinciple TheP-idealdichotomy
ThestatementofMRP
The statement of MRP
Definition
Let θ be a regular cardinal, let X be uncountable, and let
M ≺ H be countable such that [X]ω ∈ M. A subset Σ of [X]ω is
θ
M-stationary iff for all E ∈ M such that E ⊆ [X]ω is club,
Σ∩E ∩M 6= ∅.
If Σ in M then this is just the usual notion, but we will be
interested precisely in the case when Σ is not in M.
Example
LetM ≺ H . Letδ = M ∩ω andsupposeA ⊆ δ hasordertype
ℵ2 1
< δ. Then δ\A is M-stationary.
TheMappingReflectionPrinciple TheP-idealdichotomy
ThestatementofMRP
Recall the Ellentuck topology on [X]ω. Basic open sets are of
the form
[x,N] = {Y ∈ [X]ω:x ⊆ Y ⊆ N}
where N ∈ [X]ω and x ⊆ N is finite.
TheMappingReflectionPrinciple TheP-idealdichotomy
ThestatementofMRP
Definition
A set mapping Σ is open stationary iff there is an uncountable
set X = X and a regular cardinal θ = θ such that [X]ω ∈ H ,
Σ Σ θ
dom(Σ) is a club in [H ]ω and Σ(M) ⊆ [X]ω is open and
θ
M-stationary, for every M ∈ dom(Σ).
The Mapping Reflection Principle [Moore]
If Σ is an open stationary set mapping whose domain is a club,
~
there is a continuous ∈-chain N = (N :ξ < ω ) of elements
ξ 1
dom(Σ) such that for all limit ordinals ξ < ω there is ν < ξ such
1
that N ∩X ∈ Σ(N ) for all η such that ν < η < ξ.
η Σ ξ
If (N :ξ < ω ) satisfies the conclusion of MRP for Σ then it is
ξ 1
said to be a reflecting sequence for Σ
TheMappingReflectionPrinciple TheP-idealdichotomy
ThestatementofMRP
Definition
A set mapping Σ is open stationary iff there is an uncountable
set X = X and a regular cardinal θ = θ such that [X]ω ∈ H ,
Σ Σ θ
dom(Σ) is a club in [H ]ω and Σ(M) ⊆ [X]ω is open and
θ
M-stationary, for every M ∈ dom(Σ).
The Mapping Reflection Principle [Moore]
If Σ is an open stationary set mapping whose domain is a club,
~
there is a continuous ∈-chain N = (N :ξ < ω ) of elements
ξ 1
dom(Σ) such that for all limit ordinals ξ < ω there is ν < ξ such
1
that N ∩X ∈ Σ(N ) for all η such that ν < η < ξ.
η Σ ξ
If (N :ξ < ω ) satisfies the conclusion of MRP for Σ then it is
ξ 1
said to be a reflecting sequence for Σ
TheMappingReflectionPrinciple TheP-idealdichotomy
ThestatementofMRP
Definition
A set mapping Σ is open stationary iff there is an uncountable
set X = X and a regular cardinal θ = θ such that [X]ω ∈ H ,
Σ Σ θ
dom(Σ) is a club in [H ]ω and Σ(M) ⊆ [X]ω is open and
θ
M-stationary, for every M ∈ dom(Σ).
The Mapping Reflection Principle [Moore]
If Σ is an open stationary set mapping whose domain is a club,
~
there is a continuous ∈-chain N = (N :ξ < ω ) of elements
ξ 1
dom(Σ) such that for all limit ordinals ξ < ω there is ν < ξ such
1
that N ∩X ∈ Σ(N ) for all η such that ν < η < ξ.
η Σ ξ
If (N :ξ < ω ) satisfies the conclusion of MRP for Σ then it is
ξ 1
said to be a reflecting sequence for Σ