ABSTRACT
There is a temptation to cut through the foundational problems by using the nonconstructive rule of induction (the omega rule) and similar semantic concepts to characterize all true propositions in arithmetic, classical analysis, and set theory. In this way, of course, e.g. unnatural numbers are excluded by the basic principles. However, there is not much explaining left to be done, since what is to be explained is simply taken for granted. With the infinite rule, more is accepted which is a projection by analogy o f the finite into the infinite. We can never go through infinitely many steps in a calculation or use infinitely many premisses in a proof unless we have somehow succeeded in summarizing the infinitely many with a finite schema in an informative way. Both mathematical induction and transfinite induction are principles by which we make inferences after we have found by mental experimentations two suitable premisses which summarize together the infinitely many premisses needed. A very essential purpose of the mathematical activity is to devise methods by which infinity can be handled by a finite intellect. The postulation of an infinite intellect has little positive content except perhaps that it would make the whole mathematical activity un necessary.
