Mohammad Golshani (Joint work with Sy David Friedman) Institute for Research in Fundamental Sciences (IPM) Tehran – Iran 60 th Birthday of Sy David Friedman

Introduction Questions A Natural Idea Merimovich Construction Properties the Forcing Notion ℙ Must Have Solution of the Problem Open Problem

(1) Supercompact Radin forcing

(Foreman – Woodin): Starting from a supercompact cardinal 𝜅 and infinitely many inaccessibles above it, we can construct a generic extension 𝑉 ∗ ⊇ 𝑉 in which:  Cardinals are preserved,  𝜅 remains inaccessible,  ∀𝜆 < 𝜅, 2 𝜆 > 𝜆 + and in fact 2 𝜆 is weakly inaccessible.

Introduction Questions A Natural Idea Merimovich Construction Properties the Forcing Notion ℙ Must Have Solution of the Problem Open Problem

(2) Radin forcing with interleaved collapses

(Cummings): Starting from a (𝜅 + 3) a generic extension 𝑉 ∗ – strong cardinal ⊇ 𝑉 in which: 𝜅 we can construct  𝜅 remains inaccessible,  ∀𝜆 < 𝜅, 2 𝜆 2 𝜆 = 𝜆 + = 𝜆 ++ 𝑖𝑓 𝜆 𝑖𝑠 𝑎 𝑠𝑢𝑐𝑐𝑒𝑠𝑠𝑜𝑟 𝑐𝑎𝑟𝑑𝑖𝑛𝑎𝑙 𝑖𝑓 𝜆 𝑖𝑠 𝑎 𝑙𝑖𝑚𝑖𝑡 𝑐𝑎𝑟𝑑𝑖𝑛𝑎𝑙 Introduction Questions A Natural Idea Merimovich Construction Properties the Forcing Notion ℙ Must Have Solution of the Problem Open Problem

(3) Extender based Radin forcing

(Merimovich): Let 𝑛 > 1 . Starting from a (𝜅 + 𝑛 + 1) there exists a generic extension 𝑉 ∗ ⊇ 𝑉 – strong cardinal 𝜅 , in which:  𝜅 remains inaccessible,  ∀𝜆 < 𝜅, 2 𝜆 = 𝜆 +𝑛 Introduction Questions A Natural Idea Merimovich Construction Properties the Forcing Notion ℙ Must Have Solution of the Problem Open Problem

Introduction Questions A Natural Idea Merimovich Construction Properties the Forcing Notion ℙ Must Have Solution of the Problem Open Problem

Introduction Questions A Natural Idea Merimovich Construction Properties the Forcing Notion ℙ Must Have Solution of the Problem Open Problem



Question:

Is it possible to kill GCH everywhere by a cofinality preserving forcing over a model of GCH? If so, can we have a fixed finite gap in the resulting model; meaning that

2

𝜆

= 𝜆

+𝑛 for some finite

𝑛 > 1

for all

𝜆

?



Question:

Can we have a pair

(𝑉

1

, 𝑉

2

)

of models of ZFC with the same cardinals and cofinalities such that (or even

𝑉 𝑉

1 2

⊨ 𝐺𝐶𝐻 ⊨ ∀𝜆, 2

𝜆 and

𝑉

2

⊨ ∀𝜆, 2 = 𝜆

+𝑛 )?

𝜆

> 𝜆

+ Introduction Question A Natural Idea Merimovich Construction Properties the Forcing Notion ℙ Must Have Solution of the Problem Open Problem

Introduction Questions A Natural Idea Merimovich Construction Properties the Forcing Notion ℙ Must Have Solution of the Problem Open Problem

Introduction Questions A Natural Idea Merimovich Construction Properties the Forcing Notion ℙ Must Have Solution of the Problem Open Problem

Introduction Questions A Natural Idea Merimovich Construction Properties the Forcing Notion ℙ Must Have Solution of the Problem Open Problem

Introduction Questions A Natural Idea Merimovich Construction Properties the Forcing Notion ℙ Must Have Solution of the Problem Open Problem

Introduction Questions A Natural Idea Merimovich Construction Properties the Forcing Notion ℙ Must Have Solution of the Problem Open Problem

Introduction Questions A Natural Idea Merimovich Construction Properties the Forcing Notion ℙ Must Have Solution of the Problem Open Problem

Introduction Questions A Natural Idea Merimovich Construction Properties the Forcing Notion ℙ Must Have Solution of the Problem Open Problem

𝑀 Introduction Questions A Natural Idea Merimovich Construction Properties the Forcing Notion ℙ Must Have Solution of the Problem Open Problem

Introduction Questions A Natural Idea Merimovich Construction Properties the Forcing Notion ℙ Must Have Solution of the Problem Open Problem

Introduction Questions A Natural Idea Merimovich Construction Properties the Forcing Notion ℙ Must Have Solution of the Problem Open Problem

Introduction Questions A Natural Idea Merimovich Construction Properties the Forcing Notion ℙ Must Have Solution of the Problem Open Problem

Introduction Questions A Natural Idea Merimovich Construction Properties the Forcing Notion ℙ Must Have Solution of the Problem Open Problem