Intuitionistic logic and axiomatic theories

This thesis explores some properties of elementary intuitionistic theories. We focus on the following theories: the theory of equality, linear order, dense linear order, the theory of a successor function, Robinson arithmetic and the theory of rational numbers with addition; moreover, we usually deal with two dierent formulations of the theories. As for the properties, our main interest is in the following four: coincidence with the classical version of a theory, saturation, De Jongh's theorem and decidability. The thesis draws especially from the results of C. Smorynski and D. de Jongh and tries to develop them. Some results known for Heyting arithmetic are proved for other theories. We also try to answer the question of what is the eect of replacing an axiom by a dierent (classically equivalent) axiom, or which properties a \good" intuitionistic theory should have