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In their 1972 paper On the number of complete Boolean algebras Monk and Solovay showed that if $\lambda$ is an infinite cardinal, then there are $2^{2^\lambda}$ many isomorphism types of complete Boolean algebras of power $2^\lambda.$ On the other hand, by a result of Pierce, a complete Boolean algebra of infinite power $\kappa$ exists iff $\kappa=\kappa^{\aleph_0}$.

In the above mentioned paper, the following is asked:

Question. Suppose $\kappa=\kappa^{\aleph_0}$, but it is not of the form $2^\lambda$ for any $\lambda$. Are there ${2^\kappa}$ many isomorphism types of complete Boolean algebras of power $\kappa?$

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    $\begingroup$ One could also naturally ask for lower bounds on the number of isomorphism types. $\endgroup$
    – Joel David Hamkins
    Commented Feb 22 at 13:39

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The answer is that there are still $2^\kappa$ many isomorphism types of complete Boolean algebras of power $\kappa$.

This is proved by Shelah, see Building complicated index models and Boolean algebras, Conclusion 2.17.

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    $\begingroup$ Perhaps you could say something about the argument method? $\endgroup$
    – Joel David Hamkins
    Commented Feb 27 at 11:54
  • $\begingroup$ @JoelDavidHamkins The proof, as one might expect, is based on several earlier definitions and lemmas, so rather long. So I think it is easier to just look at the paper and the proof. $\endgroup$
    – Mohammad Golshani
    Commented Mar 13 at 2:33

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