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Peano Arithmetic

  Peano arithmetic is the axiomatization of the basic operations of arithmetic most commonly used today by mathematicians. It is named after its originator, Giuseppe Peano (1858 - 1932). It also acts as the basis for modern ideas about numbers.

The basic elements of Peano\'s system are the concepts of zero and successor. The successor is a function which maps a number to the number which will come after it when counting. These two concepts are the ones for which Peano actually gave axioms; the basic operations of arithmetic are then easily defined. The axioms basically say that every number has a unique successor, that every number except 0 is the successor of some other number, that no number is the successor of two distinct numbers, and, most importantly, the principle of induction. Induction is not an axiom, but what is known as an axiom schema, an infinite collection of axioms (in this case, one for every property of numbers expressible using symbolic logic and the concepts of 0 and successor). The principle of induction is that if P is a property, such that P(0) is true, and for every number n, if P(n) is true, then so is P(S(n)) (where S(n) is the successor of n); then P(m) is true for every number m.

The operations of addition (+) and multiplication (×) are defined using the principle of recursion. This is derived from the principle of induction, and says that we can define a function f by giving its value at 0 (f(0)), and its value at the successor of n in terms of its value at n, for every number n. As examples of this, the operation of ‘adding m to a’, where a is a fixed number, is defined by saying that a + 0 = a, and a + S(m) = S(a + m); and the operation of ‘multiplying a by m’ is defined by saying that a × 0 = 0, and a × S(m) = (a × m) + a (where, in both cases, S(m) is the successor of m).

With these definitions, it is easy to prove the basic truths of arithmetic, for example that 1 + 1 = 2. (This proof took Bertrand Russell hundreds pages to prove, because of his unwieldy definition of number.) It is important to remember that 1 and 2 are really just shorthand ways to write down S(0) and S(S(0)) respectively, and that neither term has any meaning in the language of logic in which Peano arithmetic operates. The proof goes as follows: S(0) + S(0) = S(S(0) + 0) = S(S(0)), which is considerably shorter than Russell\'s. SMcL



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