Measurable cardinal
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In mathematics, a measurable cardinal is a certain kind of large cardinal number.
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[edit] Measurable
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 means that 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.
Equivalently, κ is a measurable cardinal if and only if it is an uncountable cardinal with a <κ-additive, non-principal ultrafilter.
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 cardinal. 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.
[edit] Real-valued measurable
A cardinal κ is real-valued measurable means that 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 if there is a countably additive extension of the Lebesgue measure to all sets of real numbers.
Existence of measurable cardinals in ZFC, real valued measurable cardinals in ZFC, and measurable cardinals in ZF, are equiconsistent.
[edit] See also
[edit] References
- Drake, F. R. (1974). Set Theory: An Introduction to Large Cardinals (Studies in Logic and the Foundations of Mathematics ; V. 76). Elsevier Science Ltd. ISBN 0-444-10535-2.
- Kanamori, Akihiro (2003). The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings, 2nd ed, Springer. ISBN 3-540-00384-3.
