A remark on the ultrapower cardinality and the continuum problem c 1 - PowerPoint PPT Presentation

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, Serbia aljosha@lycos.com 2 Faculty of transportation and traffic engineering Vojvode Stepe 305 11000 Belgrade, Serbia pera@sf.bg.ac.yu

Regular ultrafilters Two cardinal problem Continuum displacement Outline Regular ultrafilters 1 Two cardinal problem 2 Continuum displacement 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.

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 | = 2 κ . | D

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 | = 2 κ . | D

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 , U M , . . . ) such that: M | = T | M | = κ | U M | = λ .

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

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.

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 ( | � Λ( κ ) | , | � κ | ) and D D ( | � κ | , | � Γ( κ ) | , ), so D D � � � | κ | � | Λ( κ ) | � Λ( | ( κ ) | ) D D D and � � � Γ( | κ | ) � | Γ( κ ) | � | κ | . D D D

Regular ultrafilters Two cardinal problem Continuum displacement The continuum function is a cardinal function ℵ α �→ 2 ℵ α , α ∈ On . Definition The continuum displacement function is an ordinal function f : On − → On such that 2 ℵ α = ℵ α + f ( α ) , α ∈ On .

Regular ultrafilters Two cardinal problem Continuum displacement Example Initial boundaries on the CP-displacement f : 2 ℵ α > ℵ α , so f ( α ) � 1. 2 ℵ α � ℵ 2 ℵ α = ℵ α +2 ℵ α , so f ( α ) � 2 ℵ α .

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 .

Regular ultrafilters Two cardinal problem Continuum displacement Theorem Let ( ℵ ξ ( λ ) , λ ) be LLG for theory T for all infinite cardinals λ . Fix some κ � ω . Suppose that ℵ ξ ( κ ) < ℵ ξ ( κ ) = ℵ ξ ( κ ) and that D is a uniform, nonregular ultrafilter over ℵ ξ ( κ ) “jumping” after κ , i.e. � � | κ | < | ℵ ξ ( κ ) | . D D Let ℵ α = | � D κ | and ℵ β = ℵ ξ ( κ ) . Then, α < β + f ( β ) � α + ξ � α + β.

Regular ultrafilters Two cardinal problem Continuum displacement � ℵ α = | κ | D � | ℵ β | < D 2 ℵ β = = ℵ β + f ( β ) Hence, α < β + f ( β ) .

Regular ultrafilters Two cardinal problem Continuum displacement � ℵ β + f ( β ) = | ℵ ξ ( κ ) | D � ℵ ξ ( | κ | ) � D = ℵ ξ ( ℵ α ) = ℵ α + ξ ℵ α + β . � Thus, β + f ( β ) � α + ξ � α + β.

Regular ultrafilters Two cardinal problem Continuum displacement Example Let κ = ω , ξ = 17. Then, 2 ℵ 17 � ℵ α +17 . In addition, if | � ω | � ℵ 17 , then D 2 ℵ 17 � ℵ 34 .

Regular ultrafilters Two cardinal problem Continuum displacement Example If 2 ℵ 17 = ℵ ω +1 , then there is no “jumping” ultrafilter over ℵ 17 , i.e. � � | ω | = | ℵ 17 | . D D