Intuitive Maths

NT Insights Vol 1.1 Addition on Natural Numbers We have already seen how the 5 axioms together generate the entire set of natural numbers in Vol 1.0 . Now we will define a binary operation, called addition, on in terms of those 5 axioms. Before that, let us give these elements of their usual names, as follows: and so on… An important observation is that the successive natural numbers differ by a single reference to . Let us call a reference to as a unit distance. Then, clearly, the successive natural numbers are unit distance apart. Also, the difference between any two natural numbers is a multiple of this unit distance. Another important observation is that does not involve any reference to function . This is so because all other numbers are successors of , but is not successor of any natural number (Reminds me of Lord Shiva, who is called Anaadi). The magnitude of distance, therefore, can not be assigned to . Whereas, for every non-zero natural number , it...

NT Insights Vol 1.0 Axiomatizing Natural Numbers One can not prove theorems by assuming nothing. But what is more fascinating is that a large number of concepts can be explained by a fixed number of elementary notions, known as axioms . Axioms are assumed to be self-evident truths. One such set of axioms about Natural Numbers was proposed by Peano: Peano’s Axioms Let denote a set. Suppose we are not permitted to see its contents. Instead, we are given the following information about : A mapping , different from indentity map, such that is invariant under , i.e., Pre-image of does not exist under , i.e., there does not exist an such that The mapping is injective, i.e., If with and , then With these axioms alone, we can describe/generate the entire set , without actually seeing its elements. Let us see HOW . Let us start by analyzing what are we provided with. We are sure that there exists an element denoted by in . So is non-empty. Additional...