This chapter demonstrates the informal rigor and completeness proofs in mathematical philosophy. Informal rigor makes the analysis precise to eliminate the doubtful properties of the intuitive notions. The principal emphasis is on intuitive notions, which do not occur in ordinary mathematical practice but leads to new axioms for current notions. The difference between familiar independence results and the independence of the continuum hypothesis is discussed in the chapter; the difference is formulated in terms of higher order consequence. The chapter discusses the relation between intuitive logical consequence on the one hand and so-called “semantic syntactic” consequence on the other. Brouwer's empirical propositions and proof considers a striking use of a new primitive to derive a purely mathematical assertion. The chapter explores the completeness questions for classical and intuitionist predicate logic. The problems of completeness involve informal rigor, at least when deciding completeness with respect to an intuitive notion of consequence.