An earlier, not yet conclusive, attempt at formulating a theory of this kind was made by Scott 1970. Also related, although less closely, are the type and logic free theories of constructions of Kreisel 1962 and 1965 and Goodman 1970. In its first version, the present theory was based on the strongly impredicative axiom that there is a type of all types whatsoever, which is at the same time a type and an object of that type. This axiom had to be abandoned, however, after it was shown to lead to a contradiction by Jean Yves Girard. I am very grateful to him for showing me his paradox. The change that it necessitated is so drastic that my theory no longer contains intuitionistic simple type theory as it originally did. Instead, its proof theoretic strength should be close to that of predicative analysis.

In arithmetic, the induction principle permits to give more direct and more intuitive proofs than alternative principles such as the existence of a minimum to all non-empty sets of natural numbers or the existence of a maximum to all bounded non-empty sets of natural numbers, that usually require detours in proofs. Moreover, when we prove a general property by induction and use it on a particular natural number, we can eliminate the invocation of the induction principle and get a more elementary proof [1]. The idea of induction is that when a property holds for some natural number a and is hereditary, then it holds for all numbers greater than or equal to a. The fact that the property is hereditary means that when it holds for some number n then it propagates "at least a little bit " to greater numbers. In various formulations of the scheme, this takes the form P (n) ) P (S(n)) or (8p! n (P p)) ) (P n). We present in this note a similar principle for real numbers. The fact that some property is hereditary is expressed by the fact that if P holds for a real number c, then its holds on an interval [c:::c + "]. Of course, this is not enough to prove that if P holds for a real number a, then it holds for all real numbers greater than a, because unlike a bounded set of natural numbers, a bounded set of real numbers need not have a maximum. However, a closed bounded set of real numbers does. Thus, we shall restrict our induction principle to closed properties P. In this note, we state the induction principle, we prove it and we give several examples applications. All these examples come from [3] where we have formalized and proved results in elementary calculus and kinematics to studying the motion of aircraft and where this real induction principle is implicit. We also briefly discuss which axioms would be replaced by this scheme in an axiomatization of analysis, how this this induction scheme is related to ordinal induction and how, in some cases, the invocation of this scheme can be eliminated when we apply a general theorem to a particular real number.