\(\mathbb{N}\)
There are 2 common ways to construct \(\mathbb{N}\)
- by
Peano Axioms
- 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 \(\mathbb{R}\)
- by Axioms (
Field Axioms
,Order Axioms
andCompleteness Axiom
) - by
Cauchy Sequence
- by
Dedekind Cut
Field Axioms
A1 A2 A3 A4
M1 M2 M3 M4
DL
Order Axioms
Completeness Axiom
Archimedean property
\(\mathbb{C}\)
\[ \mathbb{C} = \{a+bi \mid a, b \in \mathbb{R}, i^2 = -1 \} \]