 There are many ways to further extend second-order logic. The most obvious is third, fourth, and so on order logic. The general principle, already recognized by Tarski (1933 [1956]), is that in higher order logic one can formalize the semantics—define truth—of lower order logic. Therefore one certainly gains expressive power by using higher order logic. Let us take number theory as an example. With first order logic we can express properties such as “n is a prime number” and propositions such as “there are infinitely many prime numbers”, Fermat’s Last Theorem and Goldbach’s Conjecture. First order properties of natural numbers fall into a natural hierarchy called the arithmetical hierarchy the levels of which are denoted \(\Sigma^0_n\) and \(\Pi^0_n\). With second-order logic we can talk about properties of sets of natural numbers, or in other words, properties of real numbers. In particular, we can define the rationals. A continuous function on the real numbers is determined by its values on rationals. Also, an open set is determined by which rational points it contains. Using these observations we can talk about continuous functions and open sets of reals. Basic facts in calculus, such as Bolzano’s Theorem stating that “If a continuous function has values of opposite sign inside an interval, then it has a root in that interval” are expressible in second-order logic over the structure \((\oN,+,\cdot)\) of natural numbers. Second-order properties of natural numbers fall into a natural hierarchy called the analytical hierarchy and its levels are denoted \(\Sigma^1_n\) and \(\Pi^1_n\). With third order logic we can talk about sets of real numbers. We can write the Continuum Hypothesis in third order logic and ask the hard question, whether the model \((\oN,+,\cdot)\) satisfies this sentence. Third order properties of natural numbers have again a natural hierarchy and its levels are denoted \(\Sigma^2_n\) and \(\Pi^2_n\).