| Measurable Cardinal |
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| CATEGORIES ABOUT MEASURABLE CARDINAL | |
| large cardinals | |
| determinacy | |
| measures set theory | |
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Formally, a measurable cardinal is a Cardinal Number κ such that there exists a κ-additive, non-trivial, 0-1-valued Measure on the Power Set of κ. Equivalently, κ is measurable if it is the Critical Point of a non-trivial Elementary Embedding of the Universe ''V'' into a Transitive Class ''M''. This equivalence is due to Jerome Keisler and Dana Scott , and uses the Ultrapower construction from Model Theory . Since ''V'' is a Proper Class , a small technical problem that is not usually present when considering ultrapowers needs to be addressed, by what is now called Scott's trick. A cardinal κ is real-valued measurable iff there is an atomless κ-additive measure on the power set of κ. A real valued measurable cardinal is Weakly Mahlo . Thus, existence of real valued measurable cardinals not greater than c would imply the negation of the Continuum Hypothesis . A real valued measurable cardinal not greater than c exists iff there is a countably additive extension of the Lebesgue Measure to all sets of real numbers. A regular cardinal is measurable if there is a cardinal preserving generic extension of ''V'' in which the cardinal is singular. Although it follows from ZFC that every measurable cardinal is Inaccessible (and is Ineffable , Ramsey , etc.), it is consistent with ZF that a measurable cardinal can be a Successor . It follows from ZF + Axiom Of Determinacy that ω1 is measurable, and that every subset of ω1 contains or is disjoint from a closed and unbounded subset. Existence of measurable cardinals in ZFC, real valued measurable cardinals in ZFC, and measurable cardinals in ZF, are Equiconsistent . |
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