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Indexed squares

James Cummings, Ernest Schimmerling
2002 Israel Journal of Mathematics  
We construct models which distinguish various square principles, and show that a strengthened form of weak square holds in the Prikry model.  ...  We study some combinatorial principles intermediate between square and weak square.  ...  Prikry forcing, good matrices and weak square It is proved in [17] that after forcing with Prikry forcing at a measurable cardinal κ the weak square principle * κ holds.  ... 
doi:10.1007/bf02785851 fatcat:zxwn7e7x45hgxd6hyt77m72qxq

On the strong equality between supercompactness and strong compactness [article]

Arthur Apter, Saharon Shelah
1995 arXiv   pre-print
V[G] models ZFC + GCH in which, (a) (preservation) for kappa <= lambda regular, if V models "kappa is lambda supercompact", then V[G] models "kappa is lambda supercompact" and so that, (b) (equivalence  ...  We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals.  ...  The referee's many corrections and helpful suggestions considerably improved the presentation of the material contained herein and have been incorporated into this version of the paper.  ... 
arXiv:math/9502232v1 fatcat:4ds6vq5bt5b2vpzer37rjz5kta

Consistency Strengths of Modified Maximality Principles [article]

George Leibman
2004 arXiv   pre-print
, COLL (the forcing notions that collapse ordinals to omega), <kappa directed closed forcing notions, etc., both with and without parameter sets.  ...  The Maximality Principle MP is a scheme which states that if a sentence of the language of ZFC is true in some forcing extension V^P, and remains true in any further forcing extension of V^P, then it is  ...  Hoping that this restriction on parameters will not lead to inconsistency, we can at least show that its consistency is strictly beyond that of zfc.  ... 
arXiv:math/0406063v1 fatcat:nnysiq7k2rf2zfzzuv5q6o2hhu

Narrow coverings of omega-product spaces [article]

Randall Dougherty
1996 arXiv   pre-print
This paper considers such coverings for products of infinitely many sets (usually a product of omega copies of the same cardinal kappa).  ...  Results of Sierpinski and others have shown that certain finite-dimensional product sets can be written as unions of subsets, each of which is "narrow" in a corresponding direction; that is, each line  ...  Forcing and Narrow Coverings In this section, we will show that, at least for most κ and λ, the properties N N C(κ, λ, F σ ) and N N C(κ, λ, Borel) are preserved under forcing to add any number of Cohen  ... 
arXiv:math/9602204v1 fatcat:rrd36lit7zda3knce44quzw7ru

The covering lemma up to a Woodin cardinal [article]

William J. Mitchell, Ernest Schimmerling, John R. Steel
1997 arXiv   pre-print
Assume that kappa is a countably closed cardinal and that alpha is a successor cardinal of K with kappa < alpha < kappa^+. Then cf( alpha ) = kappa.  ...  A cardinal kappa is countably closed if mu^omega < kappa whenever mu < kappa. Assume that there is no inner model with a Woodin cardinal and that every set has a sharp. Let K be the core model.  ...  The research of the first author was partially supported by National Science Foundation grants DMS-9001027 and DMS-9306286.  ... 
arXiv:math/9702207v1 fatcat:vpgl3guagvb7xiren6v3s7fpn4