Peano will return to this same school several years later when older, and more subdued, students are learning how to create number sets using his axioms.
But what is Peano’s motive? The pamphlet describing these axioms (Arithmetices principia, nova methodo exposita, 1889) is in Latin! For the benefit of these untutored rubes taking up valuable educational space, the axioms are proffered in English with explanations by the harried professor.
P(1) There is a natural (counting) number * (the first number whichAh such confusion reigns. Is our hero really a hero? And who (or what) are the chimps? Perhaps Act II will provide the insight we desperately search for.
the students may think of as 1).
P(2) For each natural number n, there is a natural number n’ (called the successor of n and which the students may think of as n + 1).
P(3) There is no natural number n that has a successor equal to *.
P(4) If m and n are both natural numbers and m is not equal to n, then the successor of m does not equal the successor of n..
P(1) and P(2) starts the sequence of counting numbers. P(3) gives the it a beginning and P(4) makes sure that each number in the sequence has a distinct successor (otherwise 4 could be the successor to 3 and another number – maybe 2?)
To complete the axioms, there is the most perplexing of all, but oh so important if the student is going to count properly.
P(5) If A is a subset of the set of all the natural numbers and if * is in subset A and n’ is in subset A for every n in subset A, then A is the set of all the natural
(1)This is a rather lose translation of Peano’s Axioms from class notes for Intro to Abstract Math (2002).