Cofinality of the partial ordering of functions from ω1 into ω under eventual domination

Thomas Jech, Karel Prikry

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It is unknown whether there can exist a family of functions from ω1 into ω of size less than 2N1 that dominates all functions fromω1 Intoω. We show that there is no such family if the continuum is real-valued measurable, and that the existence of such a family has consequences for cardinal arithmetic, and is related to large cardinal axioms.

Original languageEnglish (US)
Pages (from-to)25-32
Number of pages8
JournalMathematical Proceedings of the Cambridge Philosophical Society
Issue number1
StatePublished - Jan 1984

Bibliographical note

Funding Information:
THEOREM 3*9 (Jensen's Covering Theorem [5]). / / L[U] does not exist then for every set X of ordinals of size Ki there exists a set YeK of sizefc^such that X £ Y. Combining Theorem 3-2, 3-9 and Lemma 3-8 we get: THEOREM 3-10. J/Cof (wwi) < 2Ni and if 2Ki is greater than the least possible value for 2K» then L[U] exists. This is Theorem C with L[U] in place of'a model with large cardinals'. The general theory of transitive models for large cardinals axioms along with corresponding covering theorems, as developed by Mitchell [8], provides the framework for the appropriate generalization of Theorem 3-10: 'L[U] exists' can be replaced by much stronger large cardinal axioms. Research supported by NSF grants MCS 78-01525 and MCS 80-20467.

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