Number Systems

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0. Preliminaries

This introductory section can be skipped it the reader feels comfortable with the following topics exaplained. A good knowledge from Logic and Set Theory is welcomed too.

Ordering Relations

A strict partial ordering $<$ on a set $S$ is a binary relation $R \subset S \times S$ has the following properties:

1. $a \not \lt a, \forall a \in S$. IRREFLEXIVITY.
2. $a \lt b \implies b \not \lt a, \forall a,b \in S$. ASYMETRY.
3. for $a,b,c \in S, a \lt b \ and \ b \lt c \implies a \lt c$ TRANSITIVITY.

Isomorphic Systems

Numbers are always fascinating entities. Are they real? mental contructs? does it matter?

1. The Natural Numbers System ($\mathbb{N},+,\cdot)$

Natural numbers, counting numbers are the basis for computation, and regarded as the only type of numbers that are really computable. They were formally defined by Peano in a set of axioms

1. $0 \in \mathbb{N}$
2. Given a number $n$, its successor $n \mapsto n'$
3. $\forall n \in \mathbb{N}, n' \not ={0}$.
4. $\forall n,m \in \mathbb{N}, m' = n' \implies m = n$
5. AXIOM OF INDUCTION: if $K \subset \mathbb{N}$ s.t.
   a. $0 \in K$
   b. $\forall n \in K, n' \in K$
   Then $K = \mathbb{N}$
   (1) Simply "initiates" the set with a base element, in this case '0'. (2) Just states the successor function. Example: a. 0 = 0' b. 0'' = 0'''. I cannot make 0' = 1 yet, because I need the sum operation for that. Note that we cannot define the successor in terms of sum, because we hadn't already defined the sum in $\mathbb{N}$. (3) This just states that '0' can't be the successot of any other number in $\mathbb{N}$. (4) This implies that the successor of two numbers cannot be equal, unless they are equal. (5) Induction is the heart of Peano's axioms. It states that the successor is also in the subset $K$; and as the iteration goes on inside K until $(K)' = K \cup {0}$

Briefly, the axioms define the natural numbers inductively, having zero as the base, and the successor function as the induction step. 0 is not the successor of any other number in the system.

Then a system ($\mathbb{N},+,\cdot) is defined as the set of natural numbers with closed sum and product operations.

Closure is the property of a binary operation such that $if a,b \in S, a \cdot b \in S$.

Operations,

• Subtraction is not defined to all numbers in $\mathbb{N}, only if $n \le m \implies n-m \le 0$.

• Division is not defined, but there is the concept of divisibility.

$\mathbb{N}$ is well-ordered, meaning the set has a minimum element, 0 in this case.