Stranger things about the cardinality of compact metric spaces without AC.

In set theory without the Axiom of Choice (AC ), we investigate the open problem of determining the possible cardinalities of compact metric spaces and bounds on the cardinality of such spaces. We establish the following surprising and unexpected results: There exists a permutation model N in which there is a crowded compact metric space ⟨ X , d ⟩ with | X | > | R | in N and, for every compact metric space ⟨ Y , ρ ⟩ in N , | Y | is comparable to | R | in N . This resolves an open problem from Keremedis and Tachtsis [9], Keremedis [8] and Keremedis, Tachtsis and Wajch [11]. Using the proof-technique of the above result, we also provide a general criterion for permutation models to satisfy the proposition ''The cardinality of a compact metric space is comparable to | R | ''. It is relatively consistent with ZF that there exist at least | R | compact metric spaces with pairwise incomparable cardinalities, each of which is incomparable with | R | . It is relatively consistent with ZF that there exists a crowded compact metric space ⟨ X , d ⟩ such that the set [ X ] < ω of finite subsets of X has no denumerable (i.e. countably infinite) subsets and | X | is incomparable with | R | ; this will yield that ''There exists a crowded compact metric space having no infinite scattered subspaces'' is relatively consistent with ZF . The latter result resolves an open problem from Keremedis, Tachtsis and Wajch [10]. For the proof, we construct a new permutation model having the required properties and then we transfer the result into ZF via the Jech–Sochor First Embedding Theorem. [ABSTRACT FROM AUTHOR]

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