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 language||English (US)|
|Number of pages||8|
|Journal||Mathematical Proceedings of the Cambridge Philosophical Society|
|State||Published - Jan 1984|
Bibliographical noteFunding Information:
THEOREM 3*9 (Jensen's Covering Theorem ). / / 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 , 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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