Strong covering without squares

Saharon Shelah

Fundamenta Mathematicae (2000)

  • Volume: 166, Issue: 1-2, page 87-107
  • ISSN: 0016-2736

Abstract

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Let W be an inner model of ZFC. Let κ be a cardinal in V. We say that κ-covering holds between V and W iff for all X ∈ V with X ⊆ ON and V ⊨ |X| < κ, there exists Y ∈ W such that X ⊆ Y ⊆ ON and V ⊨ |Y| < κ. Strong κ-covering holds between V and W iff for every structure M ∈ V for some countable first-order language whose underlying set is some ordinal λ, and every X ∈ V with X ⊆ λ and V ⊨ |X| < κ, there is Y ∈ W such that X ⊆ Y ≺ M and V ⊨ |Y| < κ.   We prove that if κ is V-regular, κ V + = κ W + , and we have both κ-covering and κ + -covering between W and V, then strong κ-covering holds. Next we show that we can drop the assumption of κ + -covering at the expense of assuming some more absoluteness of cardinals and cofinalities between W and V, and that we can drop the assumption that κ W + = κ V + and weaken the κ + -covering assumption at the expense of assuming some structural facts about W (the existence of certain square sequences).

How to cite

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Shelah, Saharon. "Strong covering without squares." Fundamenta Mathematicae 166.1-2 (2000): 87-107. <http://eudml.org/doc/212477>.

@article{Shelah2000,
abstract = {Let W be an inner model of ZFC. Let κ be a cardinal in V. We say that κ-covering holds between V and W iff for all X ∈ V with X ⊆ ON and V ⊨ |X| < κ, there exists Y ∈ W such that X ⊆ Y ⊆ ON and V ⊨ |Y| < κ. Strong κ-covering holds between V and W iff for every structure M ∈ V for some countable first-order language whose underlying set is some ordinal λ, and every X ∈ V with X ⊆ λ and V ⊨ |X| < κ, there is Y ∈ W such that X ⊆ Y ≺ M and V ⊨ |Y| < κ.   We prove that if κ is V-regular, $κ^+_V = κ^+_W$, and we have both κ-covering and $κ^+$-covering between W and V, then strong κ-covering holds. Next we show that we can drop the assumption of $κ^+$-covering at the expense of assuming some more absoluteness of cardinals and cofinalities between W and V, and that we can drop the assumption that $κ^+_W = κ^+_V$ and weaken the $κ^+$-covering assumption at the expense of assuming some structural facts about W (the existence of certain square sequences).},
author = {Shelah, Saharon},
journal = {Fundamenta Mathematicae},
keywords = {set theory; covering; strong covering lemma; pcf theory; inner model},
language = {eng},
number = {1-2},
pages = {87-107},
title = {Strong covering without squares},
url = {http://eudml.org/doc/212477},
volume = {166},
year = {2000},
}

TY - JOUR
AU - Shelah, Saharon
TI - Strong covering without squares
JO - Fundamenta Mathematicae
PY - 2000
VL - 166
IS - 1-2
SP - 87
EP - 107
AB - Let W be an inner model of ZFC. Let κ be a cardinal in V. We say that κ-covering holds between V and W iff for all X ∈ V with X ⊆ ON and V ⊨ |X| < κ, there exists Y ∈ W such that X ⊆ Y ⊆ ON and V ⊨ |Y| < κ. Strong κ-covering holds between V and W iff for every structure M ∈ V for some countable first-order language whose underlying set is some ordinal λ, and every X ∈ V with X ⊆ λ and V ⊨ |X| < κ, there is Y ∈ W such that X ⊆ Y ≺ M and V ⊨ |Y| < κ.   We prove that if κ is V-regular, $κ^+_V = κ^+_W$, and we have both κ-covering and $κ^+$-covering between W and V, then strong κ-covering holds. Next we show that we can drop the assumption of $κ^+$-covering at the expense of assuming some more absoluteness of cardinals and cofinalities between W and V, and that we can drop the assumption that $κ^+_W = κ^+_V$ and weaken the $κ^+$-covering assumption at the expense of assuming some structural facts about W (the existence of certain square sequences).
LA - eng
KW - set theory; covering; strong covering lemma; pcf theory; inner model
UR - http://eudml.org/doc/212477
ER -

References

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  1. [BuMa] M. Burke and M. Magidor, Shelah's pcf theory and its applications, Ann. Pure Appl. Logic 50 (1990) 207-254. Zbl0713.03024
  2. [Ca] T. Carlson, unpublished. 
  3. [DeJe] K. J. Devlin and R. B. Jensen, Marginalia to a theorem of Silver, in: Proc. ISILC Logic Conference (Kiel, 1974), Springer, Berlin, 1975, 115-142. 
  4. [Sh:b] S. Shelah, Proper Forcing, Springer, Berlin, 1982. 
  5. [Sh:g] S. Shelah, Cardinal Arithmetic, Clarendon Press, Oxford, 1994. 
  6. [Sh410] S. Shelah, More on cardinal arithmetic, Arch. Math. Logic 32 (1993) 399-428. Zbl0799.03052
  7. [Sh420] S. Shelah, Advances in cardinal arithmetic, in: Finite and Infinite Combinatorics in Sets and Logic, Kluwer, New York, 1993, 355-383. 
  8. [Sh598] S. Shelah, More on covering lemma, in preparation. 
  9. [Sh:E12] S. Shelah, Analytic guide, math.LO/9906022. 

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