As proved recently, the satisfaction problem for all prenex formulae in the set-theoretic Bernays—Shönfinkel—Ramsey class is semi-decidable over von Neumann's cumulative hierarchy. Here that semi-decidability result is strengthened into a decidability result for the same collection of formulae.
"The Bernays—Schönfinkel—Ramsey class for set theory: decidability." J. Symbolic Logic 77 (3) 896 - 918, September 2012. https://doi.org/10.2178/jsl/1344862166