Any references would be appreciated. Most places only address different vocabularies (e.g. a survey of arithmetical definability by Bes).
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$\begingroup$ In "twovariable arithmetic" does every sentence have an equivalent prenex form consisting of an $x$quantifier and a $y$quantifier (in some order) followed by an open formula? $\endgroup$ – Sidney Raffer Apr 3 '16 at 9:13

$\begingroup$ No. In the first place, there are only finitely many formulae of the type you are talking about because the open formula can only say how x and y stand in relation to each other and the vocabulary is finite. $\endgroup$ – Thinniyam Srinivasan Ramanatha Apr 3 '16 at 17:21

1$\begingroup$ Sorry, I should have asked "Is every sentence equivalent to a boolean combination of 2quantifier formulas in prenex form?" Also, It would be helpful if you said what you mean by "the two variable fragment of arithmetic". $\endgroup$ – Sidney Raffer Apr 3 '16 at 19:04

1$\begingroup$ @ThinniyamSrinivasanRamanatha That's true in a relational language, but not one with function symbols. There are infinitely many nonequivalent quantifierfree ("open") formulas, e.g. $x = y$, $x + x = y$, $x+x+x = y$, etc. $\endgroup$ – Alex Kruckman Apr 3 '16 at 19:32

1$\begingroup$ If we just look at sentences with two quantifiers, without allowing variables to be rebound, then the answer is positive in practice, and conjectured to be always positive  see the answer and comment of @FelipeVoloch at mathoverflow.net/a/21503/44143 $\endgroup$ – Matt F. Feb 8 at 17:42