Any polynomial of odd degree has a real root by the intermediate value theorem. There is nothing to say in the case of the factorial function, because it is evaluated directly from its definition. Finally, Gauss's paper gave a method of deriving the auxiliary polynomial and showed that if it has a root then this provides a root of the original one. Such sloppiness in the presentation can give the impression that induction is circular. Both the property q and the variable t must be given formal names, and the box clearly delimited.
Genealogical analogies can be useful, but there is a conflict of intuition about the direction.
Our recursive paradigm is analytic, and sub-processes are usually called children , so is the ultimate descendant . The rule reduces to the familiar Peano scheme , omitted prooftree environment which is also known as simple , primitive , or just mathematical induction . Section 2.7 considers induction and recursion for numbers and lists. If some property is not universally true of the natural numbers, there is a least number for which it fails. This least number principle depends on excluded middle, but with a little care such proofs can be made intuitionistic.