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Is it possible to define in existential second-order logic (ESO) the class of fields of characteristic zero? An easy compactness argument shows that the class of fields of positive characteristic is not definable in ESO, but as far as I can tell the compactness argument can not be used to establish that the class of fields of characteristic zero is not definable in ESO.

I'm also interested in any references where definability matters in ESO are discussed.

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    $\begingroup$ "There exists a nontrivial totally ordered subgroup". $\endgroup$
    – Emil Jeřábek
    Commented Jul 4 at 7:26
  • $\begingroup$ @EmilJeřábek Lol, thanks. Would still be interested in references. $\endgroup$
    – Reijo Jaakkola
    Commented Jul 4 at 7:33
  • $\begingroup$ Oh, I don't even need a subroup: "the additive group is totally orderable". $\endgroup$
    – Emil Jeřábek
    Commented Jul 4 at 7:38
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    $\begingroup$ How about "there is a set that contains $1$, is closed under addition, and does not contain $0$"? $\endgroup$
    – Andreas Blass
    Commented Jul 4 at 14:37
  • $\begingroup$ @AndreasBlass That's simple... So this is an EMSO property. $\endgroup$
    – Reijo Jaakkola
    Commented Jul 5 at 4:30

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