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Characterizing the powerset by a complete (Scott) sentence

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This paper is part II of a study on cardinals that are characterizable by a Scott sentence, continuing previous work of the author. A cardinal κ is characterized by a Scott sentence $ϕ_ {ℳ}$ if $ϕ_ {ℳ}$ has a model of size κ, but no model of size κ⁺.
The main question in this paper is the following: Are the characterizable cardinals closed under the powerset operation? We prove that if $ℵ_{β}$ is characterized by a Scott sentence, then $2^{ℵ_{β+β₁}}$ is (homogeneously) characterized by a Scott sentence, for all 0 < β₁ < ω₁. So, the answer to the above question is positive, except the case β₁ = 0 which remains open.
As a consequence we derive that if α ≤ β and $ℵ_{β}$ is characterized by a Scott sentence, then $ℵ_{α+α₁}^{ℵ_{β+β₁}}$ is (homogeneously) characterized by a Scott sentence, for all α₁ < ω₁ and 0 < β₁ < ω₁. Hence, depending on the model of ZFC, we see that the class of characterizable and homogeneously characterizable cardinals is much richer than previously known. Several open questions are mentioned at the end.
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  • Mathematics Department, University of Detroit Mercy, 4001 W. McNichols, Detroit, MI 48221, U.S.A.
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm222-2-2
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