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Regular ultrafilters Two cardinal problem Continuum displacement A remark on the ultrapower cardinality and the continuum problem c 1 and Aleksandar Perovi c 2 Aleksandar Jovanovi 1 Faculty of mathematics Studentski trg 16 11000 Belgrade,

SLIDE 1

Regular ultrafilters Two cardinal problem Continuum displacement

A remark on the ultrapower cardinality and the continuum problem

Aleksandar Jovanovi´ c1 and Aleksandar Perovi´ c2

1Faculty of mathematics

Studentski trg 16 11000 Belgrade, Serbia aljosha@lycos.com

2Faculty of transportation and traffic engineering

Vojvode Stepe 305 11000 Belgrade, Serbia pera@sf.bg.ac.yu

SLIDE 2

Regular ultrafilters Two cardinal problem Continuum displacement

Outline

1

Regular ultrafilters

2

Two cardinal problem

3

Continuum displacement

SLIDE 3

Regular ultrafilters Two cardinal problem Continuum displacement

Definition An ultrafilter D over κ is (α, β)-regular if there is E ⊆ D of cardinality β such that X = ∅ for all X ⊆ E of cardinality α. D is β-regular if it is (ω, β)-regular. D is regular if it is κ-regular.

SLIDE 4

Regular ultrafilters Two cardinal problem Continuum displacement

Suppose that κ is an infinite cardinal, D is a regular ultrafilter over κ and that ω λ κ. Then |

λ| = 2κ.

SLIDE 5

Regular ultrafilters Two cardinal problem Continuum displacement

Let D be an ultrafilter over infinite cardinal κ. If D is uniform (|X| = κ for all X ∈ D) and κ<κ = κ, then |

κ| = 2κ.

SLIDE 6

Regular ultrafilters Two cardinal problem Continuum displacement

Definition Let L be a first order language and let U be a unary predicate symbol of L. We say that an L-theory T admits pair (κ, λ) if there is an L-structure M = (M, UM, . . . ) such that: M | = T |M| = κ |UM| = λ.

SLIDE 7

Regular ultrafilters Two cardinal problem Continuum displacement

Some examples: If T admits (κ, λ), then T admits (κ′, λ′), where λ λ′ κ′ κ. If T admits (κ, λ) and D is an ultrafilter over κ, then T admits (|

D

κ|, |

D

λ|). There is a theory admitting (κ+, κ) for all κ and not admitting any (κ++, κ) for any κ. There is a theory admitting (2κ, κ) for all κ and not admitting any ((2κ)+, κ) for any κ.

SLIDE 8

Regular ultrafilters Two cardinal problem Continuum displacement

Definition Suppose that T admits (κ, λ). We say that (κ, λ) is: Left large gap (LLG), if T doesn’t admit (κ+, λ). Right large gap (RLG), if T doesn’t admit (κ, λ+). Large gap (LG), if it is both LLG and RLG.

SLIDE 9

Regular ultrafilters Two cardinal problem Continuum displacement

Example Suppose that (Λ(κ), κ) is LLG for T for all κ, and that (κ, Γ(κ)) is RLG for T for all κ. Then, T admits (|

D

Λ(κ)|, |

D

κ|) and (|

D

κ|, |

D

Γ(κ)|, ), so |

κ| |

Λ(κ)| Λ(|

(κ)|) and Γ(|

κ|) |

Γ(κ)| |

κ|.

SLIDE 10

Regular ultrafilters Two cardinal problem Continuum displacement

The continuum function is a cardinal function ℵα → 2ℵα, α ∈ On. Definition The continuum displacement function is an ordinal function f : On − → On such that 2ℵα = ℵα+f (α), α ∈ On.

SLIDE 11

Regular ultrafilters Two cardinal problem Continuum displacement

Example Initial boundaries on the CP-displacement f : 2ℵα > ℵα, so f (α) 1. 2ℵα ℵ2ℵα = ℵα+2ℵα, so f (α) 2ℵα.

SLIDE 12

Regular ultrafilters Two cardinal problem Continuum displacement

Example Suppose that the CP-displacement f is constant, i.e. (∀α ∈ On)f (α) = β for some fixed β ∈ On. Then β < ω.

SLIDE 13

Regular ultrafilters Two cardinal problem Continuum displacement

Theorem Let (ℵξ(λ), λ) be LLG for theory T for all infinite cardinals λ. Fix some κ ω. Suppose that ℵξ(κ)<ℵξ(κ) = ℵξ(κ) and that D is a uniform, nonregular ultrafilter over ℵξ(κ) “jumping” after κ, i.e. |

κ| < |

ℵξ(κ)|. Let ℵα = |

D κ| and ℵβ = ℵξ(κ). Then,

α < β + f (β) α + ξ α + β.

SLIDE 14

Regular ultrafilters Two cardinal problem Continuum displacement

ℵα = |

κ| < |

ℵβ| = 2ℵβ = ℵβ+f (β) Hence, α < β + f (β).

SLIDE 15

Regular ultrafilters Two cardinal problem Continuum displacement

ℵβ+f (β) = |

ℵξ(κ)|

ℵξ(|
D

κ|) = ℵξ(ℵα) = ℵα+ξ

ℵα+β.

Thus, β + f (β) α + ξ α + β.

SLIDE 16

Regular ultrafilters Two cardinal problem Continuum displacement

Example Let κ = ω, ξ = 17. Then, 2ℵ17 ℵα+17. In addition, if |

D

ω| ℵ17, then 2ℵ17 ℵ34.

SLIDE 17

Regular ultrafilters Two cardinal problem Continuum displacement

A remark on the ultrapower cardinality and the continuum problem

Aleksandar Jovanovi´ c1 and Aleksandar Perovi´ c2

Studentski trg 16 11000 Belgrade, Serbia aljosha@lycos.com

Vojvode Stepe 305 11000 Belgrade, Serbia pera@sf.bg.ac.yu

Outline

1

Regular ultrafilters

2

Two cardinal problem

3

Continuum displacement

Definition An ultrafilter D over κ is (α, β)-regular if there is E ⊆ D of cardinality β such that X = ∅ for all X ⊆ E of cardinality α. D is β-regular if it is (ω, β)-regular. D is regular if it is κ-regular.

Suppose that κ is an infinite cardinal, D is a regular ultrafilter over κ and that ω λ κ. Then |

λ| = 2κ.

Let D be an ultrafilter over infinite cardinal κ. If D is uniform (|X| = κ for all X ∈ D) and κ<κ = κ, then |

κ| = 2κ.

Definition Let L be a first order language and let U be a unary predicate symbol of L. We say that an L-theory T admits pair (κ, λ) if there is an L-structure M = (M, UM, . . . ) such that: M | = T |M| = κ |UM| = λ.

Some examples: If T admits (κ, λ), then T admits (κ′, λ′), where λ λ′ κ′ κ. If T admits (κ, λ) and D is an ultrafilter over κ, then T admits (|

D

κ|, |

D

λ|). There is a theory admitting (κ+, κ) for all κ and not admitting any (κ++, κ) for any κ. There is a theory admitting (2κ, κ) for all κ and not admitting any ((2κ)+, κ) for any κ.

Definition Suppose that T admits (κ, λ). We say that (κ, λ) is: Left large gap (LLG), if T doesn’t admit (κ+, λ). Right large gap (RLG), if T doesn’t admit (κ, λ+). Large gap (LG), if it is both LLG and RLG.

Example Suppose that (Λ(κ), κ) is LLG for T for all κ, and that (κ, Γ(κ)) is RLG for T for all κ. Then, T admits (|

D

Λ(κ)|, |

D

κ|) and (|

D

κ|, |

D

Γ(κ)|, ), so |

κ| |

Λ(κ)| Λ(|

(κ)|) and Γ(|

κ|) |

Γ(κ)| |

κ|.

The continuum function is a cardinal function ℵα → 2ℵα, α ∈ On. Definition The continuum displacement function is an ordinal function f : On − → On such that 2ℵα = ℵα+f (α), α ∈ On.

Example Initial boundaries on the CP-displacement f : 2ℵα > ℵα, so f (α) 1. 2ℵα ℵ2ℵα = ℵα+2ℵα, so f (α) 2ℵα.

Example Suppose that the CP-displacement f is constant, i.e. (∀α ∈ On)f (α) = β for some fixed β ∈ On. Then β < ω.

Theorem Let (ℵξ(λ), λ) be LLG for theory T for all infinite cardinals λ. Fix some κ ω. Suppose that ℵξ(κ)<ℵξ(κ) = ℵξ(κ) and that D is a uniform, nonregular ultrafilter over ℵξ(κ) “jumping” after κ, i.e. |

κ| < |

ℵξ(κ)|. Let ℵα = |

D κ| and ℵβ = ℵξ(κ). Then,

α < β + f (β) α + ξ α + β.

ℵα = |

κ| < |

ℵβ| = 2ℵβ = ℵβ+f (β) Hence, α < β + f (β).

ℵβ+f (β) = |

ℵξ(κ)|

κ|) = ℵξ(ℵα) = ℵα+ξ

Thus, β + f (β) α + ξ α + β.

Example Let κ = ω, ξ = 17. Then, 2ℵ17 ℵα+17. In addition, if |

D

ω| ℵ17, then 2ℵ17 ℵ34.

Example If 2ℵ17 = ℵω+1, then there is no “jumping” ultrafilter over ℵ17, i.e. |

ω| = |

ℵ17|.