, Volume 118, Issue 1, pp 289-315

On strong chains of uncountable functions

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Abstract

For functionsf,g1 → ω1, where ω1 is the first uncountable cardinal, we write thatf≪g if and only if {ξ ∈ ω1 :f(ξ)≥g(ξ)} is finite. We prove the consistency of the existence of a well-ordered increasing ≪-chain of length ω12, solving a problem of A. Hajnal. The methods previously developed by us involveforcing with side conditions in morasses which is a variation on Todorcevic'sforcing with models as side conditions. The paper is self-contained and requires from the reader knowledge of Kunen's textbook and some basic experience with proper forcing and elementary submodels.

Some of the research leading to this paper was supported by NSF of USA Grant DMS-9505098 held by the author at Auburn University, AL, USA, some was done while the author was visiting Ohio University at Athens, OH, USA from September 1997 till March 1998, and some was done at Universidade de São Paulo, where the author has been working since 1998. We would like to thank the set-theory groups from these universities for their hospitality.