# Strong covering without squares

Fundamenta Mathematicae (2000)

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

top

## Abstract

top
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, ${\kappa }_{V}^{+}={\kappa }_{W}^{+}$, and we have both κ-covering and ${\kappa }^{+}$-covering between W and V, then strong κ-covering holds. Next we show that we can drop the assumption of ${\kappa }^{+}$-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 ${\kappa }_{W}^{+}={\kappa }_{V}^{+}$ and weaken the ${\kappa }^{+}$-covering assumption at the expense of assuming some structural facts about W (the existence of certain square sequences).

## How to cite

top

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

top
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.

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.