Information AboutPeano Axioms |
| CATEGORIES ABOUT PEANO AXIOMS | |
| 1889 introductions | |
| mathematical axioms | |
| formal theories of arithmetic | |
|
Peano first gave his axioms in a Latin text ''Arithmetices principia, nova methodo exposita'' published in 1889 (Peano 1889), where Peano gave nine axioms, four axioms specifying the behaviour of the equality relation and five rules involving the specifically arithmetic terms for zero and successor. It is the latter five rules that are usually intended when one discusses the Peano axioms. Peano took logical principles to be given. Peano arithmetic constitutes a fundamental formalism for Arithmetic , and the Peano axioms can be used to construct many of the most important Number System s and structures of modern mathematics. Peano arithmetic raises a number of Metamathematical and Philosophical issues, primarily involving questions of Consistency and Completeness . THE AXIOMS Informally, the Peano axioms may be stated as follows:
Letting the first natural number be 1 merely requires replacing 0 with 1 in the above axioms, to no substantive effect. However, starting with 1 does change the recursive definitions of addition and multiplication given below. More formally and following Dedekind (1888), define a Dedekind-Peano structure as an ordered triple (''X'', ''x'', ''f''), satisfying the following properties:
then ''A'' = ''X''. The following diagram sums up the Peano axioms: : where each of the Iterate s ''f''(''x''), ''f''(''f''(''x'')), ''f''(''f''(''f''(''x''))), ... of ''x'' under ''f'' are distinct. BINARY OPERATIONS AND ORDERING The above axioms can be augmented by definitions of addition and multiplication over the natural numbers N, and by the usual ordering of N. To define Addition '+' recursively in terms of successor and 0, let ''a''+0 = ''a'' and ''a''+''Sb'' = ''S''(''a''+''b'') for all ''a'', ''b''. This turns ⟨N,+⟩ into a Commutative Monoid with Identity Element 0, the so-called Free Monoid with one generator. This monoid satisfies the Cancellation Property and is therefore embeddable in a Group . The smallest group containing the natural numbers is the Integers . Given this definition of addition and ''S''0 := 1, we can define successor in term of addition as follows: ''Sb'' = ''S''(''b''+0) = ''b''+''S''0 = ''b''+1; i.e. the successor of ''b'' is simply ''b''+1.
Define the usual ; every Nonempty Subset of N has a least element. This property is also shared by every member of N. PEANO ARITHMETIC Peano Arithmetic (PA) reformulates the Peano axioms as a first order theory with two binary Operation s, addition and multiplication, recursively defined as in the preceding section, and denoted by infix '+' and juxtaposition, respectively. The conceptual change is that the Axiom schema of Induction is replaced by a schema permitting induction only over Arithmetical formulae φ, whose symbols are just 0, ''S'', '+', juxtaposition, and quantified variables ranging over the Natural Number s. The new Axiom schema of Induction represents a Countably Infinite set of Axioms . The axioms of PA are: eq z] \iff z=0).
PA does not require the predicate "is a natural number" because the Universe Of Discourse of PA is just the natural numbers N. While only one explicit Existential Quantifier appears in the above axioms, three tacit quantifiers of that nature follow from the Closure of the natural numbers under successor, addition, and multiplication. EXISTENCE AND UNIQUENESS A standard construction in set theory shows the existence of a Dedekind-Peano structure. First, we define the if it is closed under the successor function, i.e. whenever ''a'' is in ''A'', ''S''(''a'') is also in ''A''. We now define:
The set N is the set of Natural Number s; it is sometimes denoted by the Greek letter ω, especially in the context of studying Ordinal Number s. The Axiom Of Infinity guarantees the existence of an inductive set, so the set N is well-defined. The ''natural number system'' (N, 0, ''S'') can be shown to satisfy the Peano axioms. Each natural number is then equal to the set of natural numbers less than it, so that
and so on. This construction is due to John Von Neumann . This is not the only possible construction of a Dedekind-Peano structure. For instance, if we assume the construction of the set N = {0, 1, 2,...} and successor function ''S'' above, we could also define ''X'' := {5, 6, 7,...}, ''x'' := 5, and ''f'' := successor function restricted to ''X''. Then this is also a Dedekind-Peano structure. : The Lambda Calculus gives another construction of the natural numbers that satisfies the Peano axioms. Two Dedekind-Peano structures (''X'', ''x'', ''f'') and (''Y'', ''y'', ''g'') are said to be ''isomorphic'' if there is a ; in this sense, there is a "unique" Dedekind-Peano structure satisfying the Peano axioms. (See the categorical discussion below.) CATEGORICAL INTERPRETATION The Peano axioms may be interpreted in the general context of Category Theory . Let US1 be the Category of pointed unary systems; i.e. US1 is the following category:
|
|
|