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Construction of number systems

Construction of number sxstems: Natural numbers

Peano Axioms

For constraction of number systems we need Peano Axioms. If we want fully understand numbers and number sets, we must learn how to constract each sets od numbers.

Now, we can write down Peano Axioms for natural numbers:

\mathbb{N} is a set with the following properties:

  • \mathbb{N} has a distinguished element which we call '1'
  • There exists a distinguished set map \sigma : \mathbb{N} \rightarrow\mathbb{N}
  • \sigma is one-to-one function (injective function)
  • There does not exit an element n \in \mathbb{N} such that \sigma(n)=1 (not surjective function)
  • Let set S \subset \mathbb{N} such that: (Principal of Induction)
    • 1 \in S
    • if n \in S, then \sigma(n) \in S.
  • Then set S is equal to set \mathbb{N}.

We call such a set, set of natural number \mathbb{N}

Now, we can define operation of addition + and multiplaying - by recursive rules.


(1) For all n \in \mathbb{N}, n+1=\sigma(n)

(2) For any n, m \in \mathbb{N}, n+\sigma(m)=\sigma (n+m)


(1) For all n \in \mathbb{N}, n \cdot 1=n

(2) For any n, m \in \mathbb{N}, n \cdot \sigma(m)=n \cdot m+n

Construction of number sxstems: Integers

Now, we can construct integers. Of course, we will use natural number to construct integers.

Consider the set S= \mathbb{N} \times \mathbb{N}, and relations (a,b)\sim (c,d) if is valid a+b=c+d. Let the set \mathbb{Z} be equivalence classes under this relation. Now we can define on set \mathbb{Z} like:

If sets A and B \in \mathbb{Z}, then sets A and B are non-empty subsets of set S and thus we may pick elements (a,b) form set A, and elements (c,d) from set B.

Now we can define an operation (addition) denoted by \oplus.

A\oplusB=[(a+c, b+c)]

Now, we want to define multiplication of integers denoted by \sim like:

A\simB=[(a \cdot c+b \cdot d, a \cdot d+b \cdot c)].

For natural numbers and integers we can write down corresponding properties.

  • Associativity of addition: a+(b+c)=(a+b)+c
  • Commutativity od addition: a+b=b+a
  • Additive inverse: a+(-a)=(-a)+a=0
  • Existing of neutral element for addition: a+0=0+a=a
  • Distributivity: a(b+c)=a \cdot b+a \cdot c
  • Associative of multiplication: a \cdot (b \cdot c)=(a \cdot b) \cdot c
  • Commutativity of multiplication: a \cdot b=b \cdot a
  • Existing of neutral element for multiplication: a \cdot 1=1 \cdot a=a