Measurable cardinals and the continuum hypothesis

Gitik

2014

Isr. J. Math.

“…By the results of [14], there must exist some normal measure µ β ∈ V R β and some term B β such that Pα "B β ∈ µ β and B β ⊆ A β ". By replacing each A β with…”

Section: Theorem 8 the Following Theories Are Equiconsistent: A) Zfc mentioning

confidence: 99%

“…Proof: Since |P| = κ and V "No cardinal η > κ is measurable", by the results of [14], V P "No cardinal η > κ is measurable" as well. We may thus assume that δ < κ and V P "δ is measurable", since V P "κ is supercompact and a limit of members of C".…”

Section: Lemma 24 V P "Any Measurable Cardinal Is Either a Member Ofmentioning

confidence: 99%

On tall cardinals and some related generalizations

Gitik

2014

Isr. J. Math.

“…Again by the results of [20], V "δ is λ strongly compact" as well. Thus, by level by level equivalence between strong compactness and supercompactness in V , V "Either δ is λ supercompact, or δ is a measurable limit of cardinals γ which are λ supercompact".…”

Section: The Proof Of Theoremmentioning

confidence: 69%

“…We must therefore have that λ < δ , for if not, then it must be true that in both V P ρ and V P ρ * Q = V P , δ is strongly compact up to δ . Since P ρ is forcing equivalent to a partial ordering having size less than δ, by the Lévy-Solovay results [20], V "δ < κ is strongly compact up to δ ". Hence, again by [3, Lemma 1.1], V "δ is strongly compact", a contradiction to the fact that by level by level equivalence between strong compactness and supercompactness in V , V "κ is both the least strongly compact and least supercompact cardinal".…”

Section: The Proof Of Theoremmentioning

confidence: 88%

“…Thus, by level by level equivalence between strong compactness and supercompactness in V , V "Either δ is λ supercompact, or δ is a measurable limit of cardinals γ which are λ supercompact". Once more using the results of [20] and the fact that forcing with Q over V Pρ adds no new bounded subsets of δ , in both V Pρ and V Pρ * Q = V P , either δ is λ supercompact, or δ is a measurable limit of cardinals γ which are λ supercompact. In either case, δ is not a witness in V P to a failure of level by level equivalence between strong compactness and supercompactness.…”

Section: The Proof Of Theoremmentioning

confidence: 99%

See 1 more Smart Citation

Indestructibility under adding Cohen subsets and level by level equivalence

Mathematical Logic Qtrly

2009

Strong Compactness and a Global Version of a Theorem of Ben-David and Magidor

MLQ - Math. Log. Quart.

2000

Starting with a model in which κ is the least inaccessible limit of cardinals δ which are δ + strongly compact, we force and construct a model in which κ remains inaccessible and in which, for every cardinal γ < κ, ✷ γ +ω fails but ✷ γ +ω ,ω holds. This generalizes a result of Ben-David and Magidor and provides an analogue in the context of strong compactness to a result of the author and Cummings in the context of supercompactness.Mathematics Subject Classification: 03E35, 03E55, 03E50, 03E10.In [3], Ben-David and Magidor proved the theorem Con(ZFC + GCH + κ is κ + supercompact) ⇒ Con(ZFC + ✷ * ℵω + ¬✷ ℵω ). In [2], the author and Henle were able to reduce the original hypotheses of BenDavid and Magidor and proved the theorem Con(ZFC + GCH + κ is κ + strongly compact) ⇒ Con(ZFC + ✷ * ℵω + ¬✷ ℵω ). Then, in [1], the author and Cummings proved a global version of the result of [3] by proving Con(ZFC + GCH + κ is κ +5 supercompact) ⇒ Con(ZFC + κ is κ +5 supercompact + For every singular cardinal λ < κ, there exists a stationary S ⊆ λ + such that if S = S α : α < β < cof(λ) is a sequence of stationary subsets of S, then S reflects simultaneously to cofinality δ for unboundedly many δ < λ + For every singular cardinal λ < κ, ✷ λ,cof(λ) ).The proof of this result uses the generalized form of Radin forcing given by Foreman and Woodin in [5].