# Is the question whether a FO formula F has a model of size k (k is a finite number) decidable?

Hi all,

can someone please tell me whether the question whether a FO formula F has a model of size k (where k is a finite number) is decidable?

Sure. Just check all the possible interpretations for all the formula's function and relation symbols over the domain $\{1,2,\dots,k\}$. –  Andreas Blass Nov 19 '12 at 11:32
Note that in Andreas's answer (and the question, too), $k$ can be replaced by $k$ or fewer elements without changing the character of decidability. Gerhard "Ask Me About System Design" Paseman, 2012.11.19 –  Gerhard Paseman Nov 19 '12 at 15:32