ABSTRACT. Bernays maintains that an axiom system is regarded not as a system of statements about a subject matter but as a system of conditions for what might be called a relational structure. Goodstein observes that when we contrast formal mathematics with intuitive mathematics we are not contrasting an image with reality, but a game played according to strict rules with a game with rules which change with the changing situation. Gödel holds that sets are in the same sense necessary to obtain a satisfactory theory of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions.