Axiom:Peano's Axioms
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Axioms
Peano's Axioms are a set of properties which can be used to serve as a basis for logical deduction of the properties of the Natural Numbers.
They are also known as the Peano Axioms, the Dedekind-Peano Axioms and the Peano Postulates.
Let $P$ be a set which fulfils the following properties:
- P1: $P \ne \varnothing$
- P2: $\exists s: P \to P$
- P3: $\forall m, n \in P: s \left({m}\right) = s \left({n}\right) \implies m = n$
- P4: $\operatorname{Im} \left({s}\right) \ne P$
- P5: $\forall A \subseteq P: \left({\exists x \in A: \neg \left({\exists y \in P: x = s \left({y}\right)}\right) \land \left({z \in A \implies s \left({z}\right) \in A}\right)}\right) \implies A = P$
These can be written in natural language as:
- P1: The set $P$ is not empty.
- P2: There exists a successor mapping $s$ from $P$ into itself. Elements in the image of $s$ are called successor elements.
- P3: The successor mapping $s$ is injective.
- P4: The successor mapping $s$ is specifically not surjective. That is, the image of $s$ is a proper subset of $P$.
- P5: For any subset $A$ of $P$ which has an element of $P$ which is the successor of no element, such that it has the successor of every number in it, is the same set as $P$.
In Non-Successor Element of Peano Axiom Schema is Unique, we see that any two elements in $P \setminus s \left({P}\right)$ are the same element.
That is, there is one and only one element in $P$ which is not a successor element.
This element is given a specific symbol. This varies, depending on the specifics of the formulation. However, $0$ is so common nowadays as to be almost universal.
Peano Structure
Such a set $P$ , together with the mapping $s$ and non-successor element $0$ as defined above, is known as a Peano structure, or a Dedekind-Peano structure.
It is not to be confused with a Peano space, which is a concept in topology.
From the above we see that a Peano structure has the following features:
Successor Mapping
The mapping $s: P \to P$, with the properties defined above, is known as the successor mapping or successor function.
The image element $s \left({x}\right)$ of an element $x$ is called the successor element or just successor of $x$.
Non-Successor Element
The element in $P$ (usually given nowadays as $0$, i.e. zero) such that:
- $\neg \left({y \in P: 0 = s \left({y}\right)}\right)$
is known as the non-successor element of $P$.
It would be nice if there were a name for this element more terse than non-successor element and more general than zero.
Westwood suggests primal element.
Principle of Induction
Axiom P5 is known as the principle of induction.
Peano's Axioms define Natural Numbers Uniquely
The most important aspect of Peano's Axioms is that they uniquely define the set of natural numbers.
That is, not only do the natural numbers satisfy Peano's Axioms, but conversely, any set that satisfies Peano's Axioms also satisfies all the properties held by the set $\N$ of Natural Numbers.
Source of Name
This entry was named for Giuseppe Peano and Richard Dedekind.
They were formulated by Peano, and were later refined by Dedekind.
Also see
According to Paul R. Halmos: Naive Set Theory (1960):
- [These] assertions ... are known as the Peano axioms; they used to be considered as the fountainhead of all mathematical knowledge.
However, as the Peano axioms can be deduced to hold for the minimal infinite successor set as defined by the Axiom of Infinity from the Zermelo-Fraenkel axioms, it has to be pointed out that they are now rarely considered as axiomatic as such. However, in their time they were groundbreaking.
Sources
- Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts (1951): Introduction $\S 4$
- Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 12$: The Peano Axioms
- Seth Warner: Modern Algebra (1965): $\S 16$
