 is valid. We know that truth of first order sentences in \((\oN,+,\cdot,0)\) is undecidable, and even non-arithmetical by results of Gödel (1931 [1986: 144–195]) and Tarski (1933 [1956]). This shows that second-order logic is not completely axiomatizable by effective means, or decidable even in the empty vocabulary. That is, there is no decidable second-order theory with an infinite model. This is in sharp contrast to first order logic where the following theories are decidable: first order logic with empty vocabulary, first order logic with monadic predicates, dense linear order, Pressburger arithmetic (first order theory of \((\oN,+,0)\)), real closed fields (first order theory of \((\oR,+,\cdot,0,1)\)), theory of algebraically closed fields, and the first order theory of \((\oC,+,\cdot,0,1)\). In second-order logic none of the respective second-order theories are decidable. However, if we restrict second-order logic to monadic second-order logic, we obtain many important decidable theories (see  §8).