# Elimination of the quantifier “there are infinitely many”

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.

• A "parted formula" is usually called a "partitioned formula". – Alex Kruckman Jun 14 '17 at 16:06
• "Partitioned formula", indeed. – Primo Petri Jun 14 '17 at 16:23
• Do you mean for all $m>n$? – user185596 Jun 15 '17 at 22:10
• Yes. Thank you. – Primo Petri Jun 16 '17 at 6:56