Squares, scales and stationary reflection

Journal of Mathematical Logic 1 (01):35-98 (2001)

Abstract
Since the work of Gödel and Cohen, which showed that Hilbert's First Problem was independent of the usual assumptions of mathematics, there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond and square discovered by Jensen. Simultaneously, attempts have been made to find suitable natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen's principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of "PCF Theory", a generalization of cardinal arithmetic that is largely determined inside ZFC. In this paper we consider interactions between these three theories in the context of singular cardinals, focusing on the various implications between square and scales, and on consistency results between relatively strong forms of square and stationary set reflection.
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DOI 10.1142/S021906130100003X
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References found in this work BETA

Aronszajn Trees and the Independence of the Transfer Property.William Mitchell - 1972 - Annals of Pure and Applied Logic 5 (1):21.
Combinatorial Principles in the Core Model for One Woodin Cardinal.Ernest Schimmerling - 1995 - Annals of Pure and Applied Logic 74 (2):153-201.
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Reflecting Stationary Sets and Successors of Singular Cardinals.Saharon Shelah - 1991 - Archive for Mathematical Logic 31 (1):25-53.

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Citations of this work BETA

Simultaneous Stationary Reflection and Square Sequences.Yair Hayut & Chris Lambie-Hanson - 2017 - Journal of Mathematical Logic 17 (2):1750010.
Aronszajn Trees and Failure of the Singular Cardinal Hypothesis.Itay Neeman - 2009 - Journal of Mathematical Logic 9 (1):139-157.
Characterization of □Κin Core Models.Ernest Schimmerling & Martin Zeman - 2004 - Journal of Mathematical Logic 4 (01):1-72.
Square in Core Models.Ernest Schimmerling & Martin Zeman - 2001 - Bulletin of Symbolic Logic 7 (3):305-314.

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