A standard homework in measure theory textbooks asks the student to prove that there are not countably infinite -algebras. The only proof that I know is via a contradiction argument which yields no estimate on the minimum cardinality of an infinite -algebra.
Given an a set of infinite cardinality , the -algebra of all co-countable subsets of is of cardinality . This example doesn't tell me whether there are -algebras of cardinality below , if I don't assume the Continuum Hypothesis.
My question is as the title says: Are there -algebras of every uncountable cardinality?
Edit: The combined answer with Stephen, Matthew proves that the cardinality of a -algebra is necessarily at least . Further, for each cardinality with uncountable cofinality, the -algebra of countable (or cocountable) subsets of a set with cardinality , is of cardinality .
What is left is whether for with are there -algebras of cardinality . (I changed the title to reflect this.)
Thanks Stephen, Matthew, Apollo, for the combined work!
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