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Some Remarks on Normal Measures and Measurable Cardinals
Author(s) -
Apter Arthur W.
Publication year - 2001
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/1521-3870(200101)47:1<35::aid-malq35>3.0.co;2-c
Subject(s) - mathematics , measure (data warehouse) , continuum hypothesis , measurable function , null set , function (biology) , pure mathematics , discrete mathematics , combinatorics , mathematical analysis , set (abstract data type) , database , evolutionary biology , computer science , bounded function , biology , programming language
Abstract We prove two theorems which in a certain sense show that the number of normal measures a measurable cardinal κ can carry is independent of a given fixed behavior of the continuum function on any set having measure 1 with respect to every normal measure over κ . First, starting with a model V ⊨ “ZFC + GCH + o ( κ ) = δ *” for δ * ≤ κ + any finite or infinite cardinal, we force and construct an inner model N ⊆ V [ G ] so that N ⊨ “ZF + (∀ δ < κ ) [DC δ ] + ¬AC κ + κ carries exactly δ * normal measures + 2 δ = δ ++ on a set having measure 1 with respect to every normal measure over κ ”. There is nothing special about 2 δ = δ here, and other stated values for the continuum function will be possible as well. Then, starting with a model V ⊨ “ZFC + GCH + κ is supercompact”, we force and construct models of AC in which, roughly speaking, regardless of the specified behavior of the continuum function below κ on any set having measure 1 with respect to every normal measure over κ , κ can in essence carry any number of normal measures δ * ≥ κ ++ .