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  (   6   5   5   )  r  e  v   i  s   i  o  n  :   2   0   0   1    1   1    1   2  m  o   d   i   f   i  e   d  :   2   0   0   1    1   1    1   2 ITERATION OF λ-COMPLETE FOR CING NOTIONS NOT COLLAPSING λ + . ANDRZEJ ROS LANOWSKI AND SAHARON SHELAH Abstract. We look for a parallel to the notion of “proper forcing” among λ-complete forcing notions not collapsing λ + . We sugges t such a definit ion and pro ve that it is prese rve d by suitabl e itera tions . 0. Introduction This work follows [Sh 587] and [Sh 667] (and see history there), but we do not rely on those papers. Our goal in this and the previous papers is to dev elop a theory parall el to “propernes s in CS iterations” for iterations with larger supports. In [Sh 587], [Sh 667] we have presented parallels to [Sh 64] and [Sh:98], whereas here we try to hav e parallels to [Sh 100], [Sh:b, Ch.III], [Sh:b, Ch.V,§5-§7] and hopefully [Sh:f, Ch.VI], [Sh:f, Ch.XVIII]. It seems too much to hope for a notion fully parallel to “proper” among λ-complete forcing notions as even for “ λ + -c.c. λ-complete ” there are prob- lems. We should also remember about ZFC limitations for possible iteration theore ms. F or example, if in the definition of the forcing notion Q ∗ in Sec- tion 3 we demand h  p ↾ e δ ⊆ h δ , then the proof fails. This ma y se em a drawback, but one should look at [Sh:f, AP, p.985, 3.6(2) and p.990, 3.9]. By it, if S ∗ = S λ + λ , and (A δ ,h δ are as in 3.4 and) we ask a success on a club, then for some h δ : δ ∈ S λ + λ we fai l. Now, if we all ow only h δ : A δ −→ 2 and we ask for “success of the uniformization” on an end segment of A δ (for all such A δ : δ ∈ S λ + λ ), then we also fail as we may code colourings with values in λ. In the first section we formulate our definitions (including prop erness over λ, see 1.3) . We believe that our main Defini tio n 1.3 is quite reasonable and appli cable. One may also define a version of it where the diamon d is “sprea d out”. The second secti on is dev oted to the proof of the preserv ation theorem, and the next one gives three (relatively easy) examples of forcing not ions fitting our sc heme. We conclude the paper wit h the discussi on of applications and variants. Key words and phrases. Forcing, iterations, not collapsing cardinals, proper. Both authors would like to thank the United States-Israel Binational Science Founda- tion for partial support. This publication has number 655 in Shelah’s list. 1

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Page 1: Andrzej Roslanowski and Saharon Shelah- Iteration of lambda-Complete Forcing Notions Not Collapsing lambda^+

8/3/2019 Andrzej Roslanowski and Saharon Shelah- Iteration of lambda-Complete Forcing Notions Not Collapsing lambda^+

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Page 2: Andrzej Roslanowski and Saharon Shelah- Iteration of lambda-Complete Forcing Notions Not Collapsing lambda^+

8/3/2019 Andrzej Roslanowski and Saharon Shelah- Iteration of lambda-Complete Forcing Notions Not Collapsing lambda^+

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Page 3: Andrzej Roslanowski and Saharon Shelah- Iteration of lambda-Complete Forcing Notions Not Collapsing lambda^+

8/3/2019 Andrzej Roslanowski and Saharon Shelah- Iteration of lambda-Complete Forcing Notions Not Collapsing lambda^+

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8/3/2019 Andrzej Roslanowski and Saharon Shelah- Iteration of lambda-Complete Forcing Notions Not Collapsing lambda^+

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8/3/2019 Andrzej Roslanowski and Saharon Shelah- Iteration of lambda-Complete Forcing Notions Not Collapsing lambda^+

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Page 11: Andrzej Roslanowski and Saharon Shelah- Iteration of lambda-Complete Forcing Notions Not Collapsing lambda^+

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Page 12: Andrzej Roslanowski and Saharon Shelah- Iteration of lambda-Complete Forcing Notions Not Collapsing lambda^+

8/3/2019 Andrzej Roslanowski and Saharon Shelah- Iteration of lambda-Complete Forcing Notions Not Collapsing lambda^+

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Page 13: Andrzej Roslanowski and Saharon Shelah- Iteration of lambda-Complete Forcing Notions Not Collapsing lambda^+

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8/3/2019 Andrzej Roslanowski and Saharon Shelah- Iteration of lambda-Complete Forcing Notions Not Collapsing lambda^+

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Saharon Shelah- Toward Classifying Unstable Theories

Saharon Shelah- More Jonsson Algebras

Saharon Shelah- Advances in Cardinal Arithmetic

Pierre Matet and Saharon Shelah- The Nonstationary Ideal on P-kappa(lambda) for lambda singular

Pierre Matet and Saharon Shelah- Positive Partition Relations for P-kappa(lambda)

Saharon Shelah- Vive La Difference III

Andrzej Roslanowski and Saharon Shelah- Sheva-Sheva-Sheva: Large Creatures

Saharon Shelah- Non Cohen Oracle C.C.C

Saharon Shelah- Polish Algebras, Shy From Freedom

Saharon Shelah- NNR Revisited

Saharon Shelah- Model Theory Without Choice? Categoricity

Saharon Shelah- Many Partition Relations Below Density

Saharon Shelah- A Partition Theorem

Alan Mekler, Andrzej Roslanowski and Saharon Shelah- On the p-Rank of Ext

Saharon Shelah- PCF Without Choice

Saharon Shelah- Black Boxes

Saharon Shelah- Properness without Elementricity

Pierre Matet, Andrzej Roslanowski and Saharon Shelah- Cofinality of the Nonstationary Ideal

Andrzej Roslanowski and Saharon Shelah- Simple forcing notions and Forcing Axioms

Saharon Shelah- On the Arrow Property

Saharon Shelah- Further Cardinal Arithmetic

Saharon Shelah- Strongly Dependent Theories

Saharon Shelah- Pseudo PCF

Saharon Shelah- Strong Dichotomy of Cardinality

Saharon Shelah- What Majority Decisions are Possible

Saharon Shelah- Middle Diamond

Andrzej Roslanowski and Saharon Shelah- More on cardinal invariants of Boolean algebras

Andrzej Roslanowski and Saharon Shelah- Forcing for hL and hd

Saharon Shelah- 0-1 Laws

Saharon Shelah- The PCF Theorem Revisited

Saharon Shelah- Nice Infinitary Logics

Rudiger Gobel and Saharon Shelah- Generalized E-Algebras via lambda-Calculus I

Andrzej Roslanowski and Saharon Shelah- Reasonably Complete Forcing Notions

Andrzej Roslanowski and Saharon Shelah- Sweet & Sour and Other Flavours of CCC Forcing Notions

Saharon Shelah- Was Sierpinski Right IV?