cardinality quantifiers

[Kennedy, Juliette]

It seems to be still unknown. Up to now the best result seems to be that consistently the extension of first order logic by the quantifiers L(Q_1, Q_2, Q_3, Q_4) is not aleph_0-compact. This is in

Shelah, Saharon<https://mathscinet.ams.org/mathscinet/search/author.html?mrauthid=160185> (IL-HEBR-IM)<https://mathscinet.ams.org/mathscinet/search/institution.html?code=IL-HEBR-IM>
The pair $(\aleph_n,\aleph_0)$ may fail $\aleph_0$-compactness. (English summary) Logic Colloquium '01, 402–433,
Lect. Notes Log., 20,<https://mathscinet.ams.org/mathscinet/search/series.html?id=4327> Assoc. Symbol. Logic, Urbana, IL, 2005.

But e.g. the case of L(Q_4) alone is open.


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