Construct N, Z, Q, R, C

\(\mathbb{N}\)

There are 2 common ways to construct \(\mathbb{N}\)

1. by Peano Axioms
2. by Set Theory (\(\emptyset\) and \(\cup\))

Dedekind–Peano Structure is the ternary tuple \((e,S,\mathbb{N})\) such that

• \(e\) is an element, \(S\) is the a function, \(\mathbb{N}\) is a set
• \(e \in \mathbb{N}\) (neutral element)
• \(\forall a \in \mathbb{N}, S(a)\in\mathbb{N}\) (successor function is \(\mathbb{N}\rightarrow\mathbb{N}\))
• \(\forall a,b\in\mathbb{N}, (S(a)=S(b) \implies a=b)\) (successor function is injection)
• \(\forall a \in \mathbb{N}, S(a)\mathbb{N}eq e\) (the range of successor function exclude \(e\), no circle, \(e\) is the first)
• \(\forall P,\{P(e) \land \forall k \in \mathbb{N},[P(k)\implies P(S(k))]\} \implies [\forall n \in \mathbb{N}, P(n)]\) (induction)

So, we can define \(0 := e, 1:=S(0),2:=S(1)=S(S(0)), ...\)

\(\mathbb{Z}\)

By equivalence classes of ordered pairs of \(N\), we can construct \(\mathbb{Z}\)

\((a, b)\) to express \(a - b\)

\[ \mathbb{Z} = \{[(a,b)] \mid a, b \in \mathbb{N} \} \]

\(\mathbb{Q}\)

\[ \mathbb{Q} = \{\frac{m}{n} \mid m, n \in \mathbb{Z} \} \]

\(\mathbb{R}\)

There are 3 common ways to construct \(\mathbb{R}\)

1. by Axioms (Field Axioms, Order Axioms and Completeness Axiom)
2. by Cauchy Sequence
3. by Dedekind Cut

Field Axioms

A1 A2 A3 A4

M1 M2 M3 M4

DL

Order Axioms

Completeness Axiom

Archimedean property

\(\mathbb{C}\)

\[ \mathbb{C} = \{a+bi \mid a, b \in \mathbb{R}, i^2 = -1 \} \]