Is there a back-and-forth condition equivalent to elimination of the $\exists^\infty$ quantifier? (The question may not have much of a sense. Then, please, argue why.)

Background. For instance, the back-and-forth condition for elimination of (classical) quantifiers reads as follows. For every partial isomorphism $f:M\to N$ and every $a\in M$ there is $N'\succeq N$ and partial isomorphism $g:M\to N'$ that extends $f$ and is defined on $a$.

A theory $T$ eliminates the $\exists^\infty$ quantifier if for every partitioned$^{*}$ formula $\psi(x,y)$ there is an $n$ such that for all $m>n$, $\ T\ \vdash\ \exists^{\ge m}y\;\psi(x,y)\leftrightarrow\exists^{\ge n}y\;\psi(x,y)$.

($^{*}$) By partitioned formula I mean a formula and a pair of tuples of variables.

  • $\begingroup$ A "parted formula" is usually called a "partitioned formula". $\endgroup$ Jun 14, 2017 at 16:06
  • $\begingroup$ "Partitioned formula", indeed. $\endgroup$ Jun 14, 2017 at 16:23
  • $\begingroup$ Do you mean for all $m>n$? $\endgroup$
    – user185596
    Jun 15, 2017 at 22:10
  • $\begingroup$ Yes. Thank you. $\endgroup$ Jun 16, 2017 at 6:56


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