Is equality-free elementary logic with Henkin quantifiers compact?

[Guillermo Badia]

Hi all,

Is it known whether first-order logic  with Henkin quantifiers but without
equality is compact? Clearly, the version of this logic with equality isn't
since one can define Dedekind infinitude, but can an analogous argument be
given without equality in the language? Many thanks!

Cheers,
Guillermo
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