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Archive for Mathematical Logic

, Volume 56, Issue 7–8, pp 849–876 | Cite as

On constructions with 2-cardinals

  • Piotr KoszmiderEmail author
Article
  • 118 Downloads

Abstract

We propose developing the theory of consequences of morasses relevant in mathematical applications in the language alternative to the usual one, replacing commonly used structures by families of sets originating with Velleman’s neat simplified morasses called 2-cardinals. The theory of related trees, gaps, colorings of pairs and forcing notions is reformulated and sketched from a unifying point of view with the focus on the applicability to constructions of mathematical structures like Boolean algebras, Banach spaces or compact spaces. The paper is dedicated to the memory of Jim Baumgartner whose seminal joint paper (Baumgartner and Shelah in Ann Pure Appl Logic 33(2):109–129, 1987) with Saharon Shelah provided a critical mass in the theory in question. A new result which we obtain as a side product is the consistency of the existence of a function \(f:[\lambda ^{++}]^2\rightarrow [\lambda ^{++}]^{\le \lambda }\) with the appropriate \(\lambda ^+\)-version of property \(\Delta \) for regular \(\lambda \ge \omega \) satisfying \(\lambda ^{<\lambda }=\lambda \).

Keywords

Morasses Property \(\Delta \) \(\rho \)-functions Models as side conditions Hausdorff gaps 

Mathematics Subject Classification

03E05 03E35 03E75 

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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Institute of Mathematics, Polish  Academy of SciencesWarszawaPoland

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